If you are being, say, at least semiformal in your approach to set theory, whether or not objects which are not sets exist depends upon the particular brand of set theory you choose. The most common contemporary set theory, ZFC, is a "pure set theory", in which every object is itself a set, so the men indeed do not form a set.
But there are other set theories which allow non set elements, or urelements (what a great name!). In particular, Quine's New Foundations with Urelements is a relatively popular such theory.
So far as I know it is towards the philosophical end of the spectrum to worry about whether sets should be allowed to contain urelements or not. The mathematical justification for this is that, using the axiom of choice, any set can be put in bijection with a von Neumann ordinal, hence a pure set. But you should be able to speak of sets of men if you want to, I suppose.
Addendum: I like Sergei Ivanov's answer. He hits the following key point: if you ask a generic mathematician whether or not an object which is not a set can be an element of a set, you will not get either "yes" or "no" as an answer, but rather an explanation of why they regard the question as being a mathematically unfruitful one. When using sets for mathematical purposes, the "nature" of the objects which comprise sets is now regarded as being completely irrelevant. This is the "structuralist" approach to mathematics, which has been clarified and taken further by the more modern categorical approach.