Un-constructable sets are not sets that you cannot imagine. You can imagine a set and propose the condition(s) for inclusion into the set but find that a contradiction or paradox (as in Russell's Paradox) occurs with at least one possible member in consideration then that set cannot be constructed.
You just note that such an un-constructable set has occurred and move on.
Un-constructable sets are just not 'true' mathematical statements that can be admitted to mathematics, in set theory.
As with Gödel's incompleteness theorem such un-constructable sets are imagined with self referencing statements/clauses/predicates,
"A set of all sets that does not contain itself."
Does such a set contain itself?
A little more involved than the statement "This statement is false." but the same old self referencing trick.
If the set is not in itself then it should be because the predicate states so, but if included then it is a set that contains itself and so should not be included.
There is no paradox here, if you accept that there can be non-sets(un-constructable set). 'Set' and 'non-set' just the yin and yang of all things. No, null set is a set. Non-set is not a set.
What do you do with non-sets? Non-sets will be dealt with in non-set theory, not set theory. As for the statement "This statement is false.", it is not a statement that can be true or false, but a non-statement that cannot be admitted into the mathematical world of statements. If Gödel's incompleteness theorem is stated as a paradox, it is not a paradox.
Don't get too dramatic about Russell's Paradox either, it is not a true paradox.
Obviously, there can be non-numbers. Together, numbers and non-numbers makes the world whole again.
