Set Theorists get around this
paradox by making a distinction between
sets and
classes.
The class of sets that do not contain themselves as elements is not
represented by a set. (Neither is the class of sets that do contain themselves, for that matter).
In certain
axiom systems, a set cannot be a member of itself; in these systems the class of sets that do not contain themselves is equal to the class of all sets (which isn't represented by a set either).
This may sound like circular logic or defining the problem out of existence with empty sophistry. It's not. If you don't understand why not, believe me for now and I hope to eventually node enough to make it clear.