This is gonna be the thread where we talk about stuff

The correct answer is that modern set theory does not allow you to have a set that contains a set. Therefore, the set of all sets is not a valid set, and the set of all sets that do not contain themselves is not a valid set.

Did some looking into this. Assuming you're talking about ZFC when you say "modern set theory," it's not that a set cannot contain a set, but rather that a set cannot contain itself. In this light, "the set of all sets" is not in fact a set, since it would contain itself. And "all sets that do not contain themselves" would be all sets (since sets do not contain themselves by definition), and therefore the set of them would be equivalent to "the set of all sets," and therefore not actually a set, as above.