The Skolem paradox is a result of the "downwards" Skolem-Löwenheim theorem, which says that any countable system of axioms has a countable model. Axiomizations of set theory (such as Zermelo-Fraenkel) thus have theorems which appear to speak about "uncountable" sets, but can be satisfied by only a (much smaller) countably infinite number of objects.
For example, the power set P(N) of the natural numbers is strictly larger than the set of natural numbers N. But the S-L theorem says that we can use a set the same size as N as a model for all sets, including N, P(N), and the infinite chain of larger and larger power sets. This seems to contradict the fact that not all infinities are equal.
Certainly by the normal interpretation of the words used in the axioms requires an uncountably infinite model.
But the paradox arises from confusing this "English" interpretation with the one imposed by a particular model.
If we adopt a particular interpretation (and thus a model), we can't immediately take it back and demand that this mapping fit different criteria.
For example, we tend to think of a particular meaning for "straight line" in Euclidean geometry, but the first four postulates can be satisfied by entities which are curved, giving rise to non-euclidean geometry. Just as there's no real paradox in non-straight "straight lines", there's no need to be concerned about
countable "uncountable sets". Applying the Skolem theorem to the real numbers gives a nonstandard interpretation in which "real numbers" are one-to-one with the natural numbers, from our normal perspective,
but not "one-to-one" with the "natural numbers" inside the model.
The paradox seems to say more about the failures of formal systems to capture human mathematical thinking than about any weakness in set theory.