'Hamming' I only know as Hamming distance (https://en.m.wikipedia.org/wiki/Hamming_distance).
If that is what you are referring to, the first thing I would do is to divide and conquer by splitting the sets of strings by length (Hamming distance isn't defined for strings of unequal length)
After that, I would handle the larger strings by deriving 3 about-equal sized string keys from them. For example, split strings of length 14 in parts of lengths 5, 5, and 4. If two of the strings have Hamming distance <= 2, one of those derived keys must be equal in both strings by the pigeonhole principle. For length 14, and assuming a ACGT alphabet (or am I showing my ignorance here?), that divides and conquers by a factor 256 or 1024. Since the naive "compare all pairs" algorithm is O(nm) for sets of size n and m, respectively, in comparisons, that should get you a speed up of a factor of at least 4 orders of magnitude.
Another approach might be to consider all possible pairs of replacements, and check whether they could make the strings equal. For example, assume the errors are that one A got replaced by a C and one T by an A. Then, do a replace all in all strings for those replacements, look for duplicates in the results, and then check the Hamming distances between the original strings that became equal after making those replacements (I would guess it isn't necessary to physically do the string replacements; adjusting the comparison function during sorting may be faster)
Again assuming a small alphabet, that should be doable, as all possible pairs of replacements isn't that many pairs.
And of course, the two can be combined.
(It would be a bit more messy to implement and would give a bit less of a speed up, but I think the gist of the above would work fine if you meant edit distance instead of Hamming distance)
Edit: an alternative could be to split each string in trigrams ("abcde" => {"abc", "bcd", "cde"}), index them with some text search engine such as Lucene, using no tokenizer, and then do a billion searches in that set. Anything with few changes will end up high in the search results there, and it is fairly easy to determine hard bounds there. At 10ms per search and 1E9 searches, that would 'only' take 1E8 seconds, or 3 years. It is easily parallelized and distributed, though.