Yes, there is such a set theory. It is ZF minus foundation plus Aczel's "anti-foundation axiom".
One reference for this system is a book-length set of lecture notes by Aczel, http://standish.stanford.edu/pdf/00000056.pdf
In this theory, the collection of all sets that contain themselves is nonempty (in fact, it is a proper class) and contains elements that are themselves infinite.
If you wanted the collection of all sets that contain themselves to itself be a set, you could simply try to restrict the anti-foundation axiom to a some particular definable set of graphs; it is usually stated for the class of all graphs.
