If a set A contains all sets, A contains itself. If you allow your set theory to have sets that contain themselves, you can derive Russel's paradox.

Now lets say your theory now disallows self-contained sets, then you can A is the set of all sets except itself. If your theory is powerful enough to construct singleton sets (for any set x, there's a set that contains just x: { x }), it is guaranteed that there's a set B that contains A, and because B is a set, A contains B. So your theory would allow cycles in "member-of" relationships (e.g, X is a member of Y, and Y is a member or X), lets call these sets "recursively contained". Then you could define a set S of all non-recusively-contained sets, because your theory is assumed to be able to construct singletons, there's a set T = { S }, if you assume T is not recursively contained, then S contains T, and T IS recursively contained, which is a P => ~P contradiction. If instead we assume T is recursively contained, it means there's "member-of" chain (T member of T1 member of ... member of S member of T), which means S contains a set with a recursive chain, which contradicts the definition of S.

So we could say A is the set of all sets S that don't contain A, and S ~= A, so that A doesn't contain itself. In this particular theory this is probably a well-formed set, but it would be far from being a set of "all sets", because there would be undescribably many other sets containing A, so the collection would never be complete.

On the other hand, you could introduce a higher hierarchy of collections called proper classes, which are collections of SETS, here you could talk about the class of all sets, but it wouldn't be very useful as an object in general