Let $G$ be a Lie group and $M$ a manifold.
Question: Is the class of principal $G$-bundles over $M$ a set?
This question came up when I was thinking about the classifying stack of $G$. It is the category of principal $G$-bundles. The classifying stack is just a locally small category, because every manifold admits a principal $G$-bundle over the manifold. But this idea does not work if we fix a base manifold.
Without even understanding the terms, no. Unless you bound all the objects to be subsets of a particular set (e.g. $\mathcal{P(P(P(}M\times G)))$ or something), there's no reason for the collection of all sets which can be endowed with certain structure to be a set (unless no set can be endowed with such structure).
If any non-empty set can be given a certain structure, any other set of its cardinality can be given the same structure. This is known as "transport of structure".
Sometimes, if you're lucky, you can find some set $X$ which includes representatives of all equivalence class under some similarity relation (isomorphism, homeomorphism, isometric homeomorphism, what have you). But this is true if and only if you can find an upper bound of the cardinality of your objects.
For example, there is no "set of all finite groups", but since every finite groups can be realized as a set of integers, we can think of finite groups as sets of integers with some operation.