It is true that we cannot use an arbitrary property$^1$ $P$ 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 ...". Per Separation, we get that your collection of reals $A$ is in fact a set. And we may now take its supremum.
Note that this illustrates an important point about how $\mathsf{ZFC}$ (and its variants) get around Russell's paradox:
It's size,$^{2}$ not complexity of definition, which controls whether or not a collection is a set or a proper class in $\mathsf{ZFC}$.$^{3}$
Part of the success of $\mathsf{ZFC}$ is due to the ease with which we can in fact verify that something is a set. The only time you'll run into trouble is when you want to form a set which isn't a priori part of some bigger thing you already know is a set; here we may have to think a bit (although the axiom (scheme) of replacement similarly makes things usually very easy, once it's mastered).
EDIT: Per the comments below, let me sketch how to define "complete metric space" in the language of set theory. As you'll see, even the sketch is quite lengthy; if there's a particular point you'd like further information on, I suggest asking a separate question at MSE.
Here's the sequence of definitions we need to whip up:
We need to talk about ordered pairs, functions, and Cartesian products.
We need to build $\mathbb{N}$, so that we can build $\mathbb{Q}_{\ge 0}$, so that we can build $\mathbb{R}_{\ge 0}$; along the way we'll need the notions of equivalence relation and equivalence class, of course.
While the previous two points will be enough to define metric spaces ("An ordered pair $(X,\delta)$ where $X$ is a set and $\delta:X^2\rightarrow\mathbb{R}$ such that [stuff]"), to define complete metric spaces we'll also need the notions of infinite sequence and equivalence relation/class.
The first bulletpoint is standard set-theoretic fare which you'll see treated in the beginning of any text on set theory, so I'll skip it; if you're interested, though, you can start with the wiki page on ordered pairs.
The third is really the first in disguise: an infinite sequence is just a function with domain $\mathbb{N}$.
So all the "meat" is in bulletpoint $2$. We proceed as follows:
First, we'll use the von Neumann approach to $\mathbb{N}$: an ordinal is a hereditarily transitive set, ordinals are ordered by $\in$, and the finite ordinals are the ordinals which do not contain any (nonempty) limit ordinal. We then identify $\mathbb{N}$ with the finite ordinals - more jargonily, $\mathbb{N}=\omega$. We define addition and multiplication of ordinals via transfinite recursion as usual.
Next, we consider the equivalence relation $\sim$ on $\omega^2$ as follows: $$\langle a,b\rangle\sim\langle c,d\rangle \iff ad=bc,$$ and we let $\mathbb{Q}_{\ge0}$ be the set of $\sim$-classes. We lift the ordering on $\omega$ to $\mathbb{Q}_{\ge 0}$ in the obvious way.
Now we're ready to define $\mathbb{R}_{\ge 0}$, via Dedekind cuts: an element of $\mathbb{R}_{\ge 0}$ is a nonempty, downwards-closed, bounded-above subset of $\mathbb{Q}_{\ge0}$. The ordering on $\mathbb{R}_{\ge 0}$ is just $\subseteq$.
With all this in hand, the naive definitions of metric space, Cauchy sequence, and complete metric space translate into the language of set theory directly (if tediously).
$^1$Really I mean "first-order formula," but I don't want to get too much into the details.
$^{2}$Specifically, in a precise sense we have: a class is a proper class iff it surjects onto the class of ordinals. This is not the same as the principle of limitation of size, but it's of similar flavor.
$^3$I should observe that this isn't the only possible response to the need to distinguish between sets and proper classes: there are other set theories (e.g. $\mathsf{NF}$, $\mathsf{GPK^+_\infty}$, ...) which take the other approach. However, these theories make it harder to check whether something is in fact a set.