Do repositories of translated papers exist?
I am not aware of any online repository for translations or even citations to translations, though those certainly are good ideas. The closest existing option would be a translation index, which I discuss below in #4.

For my recent research I've relied on a fair amount of papers written in languages unfamiliar to me (mainly Russian and German), and the following procedure is roughly what I do when I identify a foreign language paper I want to find an English translation of. I figured this out primarily through trial and error because there were many Russian articles I really wanted to read. You may be able to use similar procedures to identify translations to a language other than English as well, though I have not.


Try searching Google Scholar for the author and a rough translation of the article title. Sometimes you can find either a direct link to a translation online or citation information for a translation in print. (This covers a number of databases with a lot of translations too, like the NASA technical reports search and DTIC.)
Check if a regular (cover-to-cover or partial) translation of the journal exists. By now I'm very familiar with the regularly translated journals in my field, so I don't need to identify these any longer. Here are some ideas about how to identify these journals:


Try a Google search for the foreign journal name. This often returns a publisher's website for the journal and/or Wikipedia, either of which often mention a translation journal.
Try checking the catalog of the Library of Congress for the foreign journal, which often mentions the existence of a translation journal. Searching WorldCat for the foreign journal name often can return a translation journal. Also, it's worth looking for collected works of an author on WorldCat, as they'll often include translated articles. And as user1310503 mentions, frequently you won't know the collected works book contains the article you want until you look at the book.

Try searching WorldCat and NTRL for the author and/or approximate translated title of the paper you are looking for. NTRL is produced by the US Department of Commerce, and it lists many translations produced by the US government. It's worth trying variations of the title and/or author's name, especially if the original was not written in the Latin alphabet as transliteration schemes can vary. If you can find the item on WorldCat or NTRL then ask a librarian about obtaining the document (or, if you're familiar with services your library offers, ask the ILL folks or whoever directly). Often you can only get citation information this way, but that still can be helpful.
Citation information for translations produced is often compiled in indices. Typically you look up a journal to see a list of articles translated from that journal. Most translation indices cover translations produced in a certain time period (note the difference with when the original article was written; I can think of an article that was translated about 25 years after the original). Some other translation indices are comprehensive up through a certain date. Below are several which come to mind immediately. Your local library may have one of these or a different series. There is much overlap between these indices, but not enough that you should only check one.


Consolidated index of translations into English. This is typically the first index I check as it is fairly comprehensive. However, it only has translations up through the 1960s in volume 1 and 1980s in volume 2. (In the case of the Smoluchowski paper the original poster mentioned, it's not listed in volume 1. Volume 2 lists translation number 69-16866-07D. As for how to obtain that, your guess is about as good as mine. I asked the British Library and they don't have it.)
World index of scientific translations.
Technical translations. Available online.
Consolidated translation survey. Available online.
Asking the British Library also can be helpful. They seem quite adept at searching through these indices (including their own, e.g., the NLL translations bulletin) and they have a large translation collection themselves. I have emailed them before asking about translations of papers I've had no luck identifying translations for, and been very pleasantly surprised by the results. They have many translations which do not appear to have been indexed publicly, including ones not indexed in their NLL translations bulletin. Obtaining copies of these translations can be rather difficult, though, as they are not always willing to send them via interlibrary loan.
The CIA FOIA Electronic Reading Room is a surprisingly good source for citations to translations and also English language abstracts for articles published in the 1940s to 1960s. The CIA seems to have scanned some sort of internal card catalog which had citations for a large amount of scientific journal articles not written in English. Often these citations include information about available translations and English abstracts. Unfortunately the card catalog is often illegible due to the poor quality of the scans. And the search has fairly limited capabilities. Still, I find this is one of the most valuable sources I've checked for translations of articles in the relevant time period simply because it often lists things which no other index has. (Not to mention the translations the CIA made themselves, some of which you can find on the website as is. I also filed a FOIA request for one CIA translation last year and am still waiting for them to finish the request.)
There are also many many other translation indices available. The above was just what comes to mind immediately. Check out some of the links I give at the bottom for more. Additionally, you can often find indices for particular series, and it's not uncommon that these series are not included in the consolidated indices I mention above.

Searching Google Books or HathiTrust can be useful to find citations of translations. Often you'll find the translation indices mentioned above, but not always as for some of the indices, the information provided is rather terse, not including even an author name or title (just the page range, journal year, journal vol., and journal issue!). Try searching for variations of the first author's last name (try different transliterations if necessary), the journal name, the year, and the first page number for the original article. Adding an English translation of the article title may also help. Try both putting the titles translation in quotes and not in quotes.
Some journals specialize in translating selected articles from foreign journals. Presumably these translations are made by request, though I am not certain. One example from my field is Fluid mechanics: Soviet research. The journal appears to now be called the 
International Journal of Fluid Mechanics Research
. I've browsed the previous title in the stacks before to look for interesting foreign research, and you might find doing the same for a similar journal in your field to be worthwhile.
Checking articles which cite the article you are interested in. Some of them might mention a translation. Also worth checking are bibliographies for your field. I can think of one translation of a very highly cited article in my field that I was entirely unaware of until I checked a bibliography for my field. (Incidentally, I have never seen any paper cite this translation, or any of the other two translations I identified for this article. On that note, you might find some redundancy in translations.)


Word of mouth can help too. I would not have heard about an entire series of translations had I not emailed some retired researchers about obtaining a copy of a particular foreign paper. To the best of my knowledge this series never appeared in any translation index, and while there is an index of them, it's not obvious how to obtain them. Thankfully one of the researchers I contacted had many of them in a storage unit and he sent them to a university library archive.

If the above fails, you can try checking abstract journals, both in your native language and foreign languages. These depend on your field. Due to the shorter length, it's often fairly trivial to translate a foreign language abstract of an article via Google Translate (few sentences vs. frequently 10+ pages). Frequently foreign abstract journals are translated or partially translated as well, sometimes under a similar name as the original journal, but other times published in an English language journal. For example, old (50s-60s) editions of Applied Mechanics Reviews sometimes have English translations of selected abstracts from different editions of Referativny Zhurnal. For a while the mechanics edition of that Russian review journal was translated as Soviet Abstracts: Mechanics, too.

Another method I've found useful to get some information about a foreign paper is to compile a list of quotes about the paper from articles that cite it. This will often summarize the most important aspects of the paper in a language you understand. Unfortunately, there are some disadvantages. One is that a paper may be commonly cited for one topic, when it actually discusses several topics which are of interest to you. You may never learn of the topics not mentioned in the citations this way. It's also not uncommon for an author to cite a paper without reading it, so some of what is said will be merely rehashing of what others said. The "telephone game"/"Chinese whispers" aspect can come into play as well; I can recall one particularly hard to find paper I tracked down which said nothing at all about a certain subject on which it's commonly cited for. So this may not be entirely reliable.

What can you do if you've identified a citation for a translation but can't find a copy of the actual document anywhere? If you're unfamiliar with obtaining documents from other libraries, ask a librarian as they can help you. You can also search for the translation series on WorldCat and see if you can identify libraries with that series. Often it's possible to buy a translation listed on NTRL (though not always). Sometimes visiting a library with large translation collections can help too. The Library of Congress's technical reports section has proved very helpful to me in this case. Unfortunately, you will not always be successful even if you find clear citation information. It's not uncommon to find out that a translation existed at one point but is now unavailable or inaccessible. Reasons a translation could now be unavailable include it being thrown away, it being misplaced, it being destroyed in a fire, etc. Some libraries do not participate in interlibrary loan, so you may not be able to get copies of anything from those places. A translation could also seem unavailable if few libraries which have translations from a particular series have the one you are interested in. You might have to contact many libraries before finding one with the particular document you want.

After that point, if you really want a translation of the article but have not found one, you could try paying a translator or using Google Translate. I've translated several articles into English via Google Translate now, and while it is tedious and the result often is not ideal, I do find it to be worthwhile for certain articles. As an example, here's a series of three 1938 articles which I translated from Russian and posted to my university's repository.

I am less knowledgeable about finding translations of books. I check WorldCat for the authors of a book I'm interested and sometimes am successful. Sometimes you may find translations of isolated chapters listed in the translation indices. For translations of books to English, contacting the British Library may also be useful as I recall they had a special program for this at one point.

This is a rather large topic, actually, and I certainly have missed a lot, so feel free to ask any questions and I'll do my best to answer them.

Other guides with additional information about obtaining translations:


"How to obtain a translation" by Betty B. Brociner (1977) (Dated, but has a lot of information about how to find translations and also how to request a translation from a translator.)
"Translations of scientific and technical literature: a guide to their location" by Kathleen Wright (1987) (Newer but still dated.)
Caltech
Cornell, another at Cornell
Linda Hall Library