There are alternatives such as type theory. You could also simply consider the formal first-order theory of categories itself and its extensions without considering (set-theoretic) models. If you wanna get real fancy, you can consider models of that theory in any category with finite limits which leads to the notion of an internal category. From this last perspective, the category of ($V$-small) sets is but one category that has finite limits. Internal categories in the category of ($V$-small) sets are categories.
It's worth noting that the set-theoretic foundations presented in Categories for the Working Mathematician already go beyond ZFC set theory. If I remember correctly, the set theory sketched is essentially Tarski-Grothendieck set theory (though it may just assume some Grothendieck universes and not something equivalent to Tarski's axiom).