Russell's Paradox did not call into question the Law of the Excluded Middle. In the early 1900's, it called into question an early axiomatization of Cantor's set theory by G. Frege. These axioms were shown to be inconsistent in that it was possible, using Frege's axioms, to both prove and disprove that there exists a set of those and only those sets that are not elements of themselves.
$$\exists k: \forall a: [a\in k \iff a\notin a] \land \neg\exists k: \forall a: [a\in k \iff a\notin a]$$
Various patches were put forward to make it impossible to derive this contradiction, the most popular solution being the Zermelo-Fraenkel Axioms of Set Theory. They, too, are based on classical logic including the Law of the Exclude Middle.
Note that, by using the ordinary rules of logic (including LEM) and any binary predicate $R$ (not just $\in$) we still have
$$\neg\exists: k:\forall a:[R(a,k) \iff\neg R(a,a)]$$
No contradications arise here.