It is true that we cannot use an arbitrary property$^*$ $P$ to define a set, in the sense that the collection of all things with property $P$ need not be a set. However, the axiom (scheme) of separation says that we can use an arbitrary property to define a subset: whenever $X$ is a set, the collection $\{x\in X: P(x)\}$ is also a set.
So just take $X=\mathbb{R}$ and $P(x)$ = "There is a Hilbert space such that ...".
$^*$Really I mean "first-order formula," but I don't want to get too much into the details.