Beyond Set Theory

The following is a statement from Categories for the Working Mathematician by Saunders Mac Lane (page 10) :

A category will mean any interpretation of the category axioms within set theory.

Will someone kindly explain what the author means by the phrase 'within set theory'? Is there something beyond or in the complement of it. Thank you for your help.

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.