In particular I was thinking of undistinguished countable sets (although I was confusingly throwing in ordering of the integers to try to make my point more accessible) that you then add topological structure on top of.

In that world the integers are simply the discrete topology on a countable set. Or more explicitly (to contrast with the next definition), where all singleton sets are open.

The rationals then are formed by any metric whose induced topology does not include singleton sets.

That is, any attempt to uniformly bring elements "closer" than the world where single points are open gives rise to a topology homeomorphic to the rational numbers.