Gerald's answer is quite correct.  This began as a comment justifying it, but because of length considerations I'm leaving it as an answer instead.

A model of ZFC is <b>not</b> a perfect replica of the category of sets crammed into a single set (in fact, to the extent that such a statement makes any sense at all, there is no such thing).  Rather it is a set $M$ together with a binary relation, $\in$, such that the axioms of ZFC set theory -- i.e., a certain family of first order statement in the (countable) language of sets -- holds in $(M,\in)$.  

There are a lot of things that such a model $M$ will not tell you about the category of sets.  However, assuming -- as we generally do -- that there really is a category of sets satisfying each of the axioms of ZFC, then it follows from Godel's Completeness Theorem that it must have a model $M$ as above.  This is a nontrivial result.  Moreover, since the language of sets is countable it follows from Skolem-Lowenheim that models of ZFC exist of all infinite cardinalities.  

To be fair, Skolem himself found this consequence -- the existence of a countable model of set theory -- to be somehow problematic ("Skolem's Paradox").  But modern set theorists and logicians simply don't feel this way: they have gotten used to the statement, which is not paradoxical in the strict logical sense (there is no contradiction) but rather merely sounds strange when you first hear it.

In much the same way, Tarski's motivation for the Banach-Tarski paradox was to exhibit the absurdity of the Axiom of Choice (AC).  But nowadays 99.9% of all mathematicians are happy with AC, and the study of "paradoxical decompositions" is a flourishing subfield of geometric group theory. 

As Joel David Hamkins explained to me previously here on MO, you could consider class-valued models of a theory and, depending upon taste, it may sometimes be convenient to do so.  But again it is not necessary to do so because of Godel Completeness.