Large categories? How is a set of objects/arrows not a set?

A set of objects for the category of all sets would be a set of all sets--or at least a set of all ordinals. Such a monster violates the axiom of foundation which is part of ZF. The usual solution--introduced by MacLane--is to work with a set $V$ of 'small sets' which is a model of ZF instead. This is justified because if ZF is consistent, then there is a model that provides such a set of small sets by Gödel's completeness theorem. This set $V$ defines a category of small sets, on top of which mathematicians can construct other categories of small objects. By contrast with the small sets inside of $V$, the sets outside of $V$ are 'large'. So a large category is simply a category whose set of objects is large, like the category of all small sets.