Large categories? How is a set of objects/arrows not a set? Good night,
I found there are categories whose objects and arrows aren't sets, called large categories. I understand if each object isn't a set, but the set of all objects should be a set, even if its an infinite one. Is this because of the way a set is defined? I simply can't understand this; I didn't find anything that could clear this, neither here on Wikipedia and other search results in Google (although I must say, I'm not a good googler).
Thanks for understanding, I hope this isn't a stupid question.
 A: 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.
A: As you may know, because of Russel's Paradox that appears in naive set theroy, in ZF (a standard set theory) no set can countain itself; and "$\text{Sets}$", the category of all sets, could not exist with a set of objects.
Thus, we need a different type of object collection, with different rules: a class. ZF doesn't axiomatize classes, they're informal notions, kind of workarounds. But even in ZF they do solve the problem: you can have a class of all sets; this class isn't a set, and as such, it is called a proper class.
For instance, the collection of objects (and, naturally, of arrows) in "$\text{Sets}$" category is a proper class. Categories like this are known as large categories, unlike small categories, whose collections of objects/morphisms are sets. It's not a sea of roses though, classes have other limitations (varying from theory to theory) that sets don't (otherwise, as you might have been thinking, Russel's Paradox would apply to them too, and you'd need multiple types of classes). 
Directly answer your question: in no way; it is a proper class. About "Is this because of the way a set is defined?": yes, it is.
