Could completely public passphrase hashes ever be reliably secure? This is a hypothetical question and I only have a basic understanding of Cryptography.

If one were to follow the very best cryptographic practices for storing passphrases, could it ever be possible to publish literally all information (the usernames, the hashes, the salts, the algorithms used) and still be reasonably certain that no-one could crack them?

I'm assuming there would have to be a minimum-length constraint - like 12, or maybe 16 characters minimum - and obviously if someone were to choose "1111111111111111" then it would be easily guessable, but assuming people choose sensible passphrases, like "correct horse battery staple", could this work?
For any value $x$ chosen randomly in a set of size $N$, and hash function $h$, publishing $h(x)$ allows for an exhaustive search on $x$ with average cost $N/2$. This is unavoidable.

The problem with passwords is that, by virtue of fitting in the brain of a human, they tend to come for a set of potential passwords of relatively small size $N$. We try to cope with that by making $h$ a slow function, and using a new variant of $h$ every time (that's what salts are about). See this answer for details on password hashing.

Now that's just an average. Though most humans will have trouble remembering many fat random passwords, a normal human can remember a really random password with enough entropy to defeat exhaustive search. This will imply some effort, so a lot of users won't do it.

Entropy is the key word. An entropy of $n$ bits means a set of passwords of size $N = 2^n$ (that's a simplification: I am assuming that all potential passwords have an equal probability of being selected). In the case of the famous comic which describes the "correct horse" method, an entropy of 44 bits is claimed. This is mathematically correct (see that answer for an analysis). However, 44 bits are not a lot, certainly not enough to defeat exhaustive search. Cryptographers have long required at least 80 bits, as a rule of thumb for "unbreakable through brute force"; and the relentless progress in technology, as well as an aesthetic appeal for powers of two, now make "128 bits" to golden rule. It can be argued that 128 bits of entropy ought to resist brute force for quite some time.

Basically, the "correct horse" password generation method assumes a list of 2048 words, and four words are selected randomly and uniformly in that list. That is, there are four successive selections of one word in the list, and the selections are independent of each other (so the same word can be selected several times). Make that twelve words instead of merely four, and your password will be rock solid, even if you publish the hash of it.

(If you use slow hashing with a salt and enough iterations, these twelve words can be reduced to, say, nine or ten words.)