<p>Let X be the set of all sets that do not contain themselves. Is X a member of X?</p>

The question is ill-posed in the standard
ZFC
(Zermelo-Fraenkel + axiom of Choice) set theory because the object thus defined is not a set.
Since (again, assuming standard ZFC) your
class
{x : x\not\in x} is not a set, the answer becomes no, it's not an element of itself (even as a class) since only sets can be elements of classes or sets.
By the way, as soon as you agree to the
axiom of foundation
, no set can be an element of itself.
Of course the nice thing about math is you can choose whichever axioms you want :) but believing in paradoxes is just weird.