Examples of sets which are not obviously sets

In my (limited) experience, it is usually easy to see when something is large enough to be a proper class, by constructing an element of the class for every set.

However, sometimes such a proper class has lots of redundant information, so we consider it modulo some equivalence relation. This way, in some cases, we end up with something small enough to be (represented by) an actual set!

Motivating example: The Witt group $W(F)$

(actually, we don't care about the group structure, I'm sorry)

Let $F$ be a field. Let $$ W(F) := \{(V, q)\ \text{quadratic}\ F\!-\!\text{vector spaces}\}/\sim $$ Where $(V, q)\sim (W, r)$ if there exist metabolic (i.e., hyperbolic plus degenerate) spaces $E_1, E_2$ such that $(V, q)\perp E_1 \simeq (W, r)\perp E_2$, where $(V, q)\perp (W,r) = (V\oplus W, (v,w)\mapsto q(v)+r(w))$.

Then $W(F)$ is a set has a set of representatives.

When I first learned this, this was not at all obvious to me.

(To be honest, I only vaguely recall this, and am not too sure of the technical details. The gist of this was however, that we only consider finite-dimensional vector spaces in the first place, which allow only for a finite number of non-equivalent quadratic forms, and even after that, most of those can be seen to be equivalent to a quadratic space of lower dimension by adding elements to make a large portion look hyperbolic)

Question

So, in a similar spirit to this question:

What are other non-obvious examples of sets, e.g. proper classes being “quotiented” to a set?

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