Safari giving "kCFErrorDomainCFNetwork error 303" when visiting a site
OK, this solution is the strangest thing. I found the solution on a Russian site, but now that the search terms for the solution are known you can find at some other sites.

The problem is a result of Safari holding too much local data for the site in question and failing to deal with that correctly. I use archive.org a lot so I had lots of LocalStorage and Cookie data for that site. The fact that this data is only in Safari, and none of my other browsers explains why they could browse the site just fine.

So, the solution: 


Go into Safari &gt; Preferences &gt; Privacy
Search for "archive.org" 
Click Remove
The problem goes away


Here's a video showing the problem before and after: https://imgur.com/gallery/d1P1FCi



I've reported this to Apple with full logs as radar #14814758.