What is the paradox in a set containing itself?

There is no paradox. In standard set theory, we have an axiom that says a set can't be a member of itself. This is the Axiom of Foundation. read more

The original Russell's paradox with "contain": The container (Set) that contains all (containers) that don't contain themselves. The Grelling–Nelson paradox with "describer": The describer (word) that describes all words, that don't describe themselves. read more

A set containing itself as an element is by definition not well-founded. Russel's paradox is a paradox in naive set theory, so without anything like the axiom of foundation. There are axiomatizations on set theory that allow regularity to be violated. read more