To me, irrationality is where you have something that is semantically inconsistent or contradictory (like a round square). Saying triangles are irrational, is irrational (precisely because triangles is not an absurd concept like round-square). Similarly rejecting a list of all lists, or a set of all sets, is irrational in my opinion because it implies that you think such a thing to be semantically inconsistent or contradictory (when it is not).
Call absolutely any thing (number, shape, tree, human, dream, colour) an ‘existent’. Call the set of all existents, ‘Existence’. Note that I am not referring to how real something is/exists, just that it is an existent (a member of Existence). Numbers are numbers (which is the same as saying numbers exist as numbers in Existence). The alternative is to say numbers don’t exist in Existence, or that there is/exists no such thing or existent as numbers, or that numbers are in non-Existence (like round squares and other absurdities).
All existents (including Existence) are a member of Existence (because they are all existents). Only Existence is a member of itself as an existent. Since no other existent is a member of itself as an existent, the set of all existents that are not members of themselves, is Existence.
Existence is a meaning, so it is a member of the set of all meanings. But then again, Existence IS the set of all meanings because there is no other thing, existent, set, or meaning that existentially contains all meanings, sets, or existents. The set of all ducks is not some existing animal or shape. The set of all ducks is Existence Itself (which is an existing meaning/set/existent/truth).