I know that you can define function spaces, and random variables are essentially functions. But from my limited knowledge, I have never ever came across the notion of the "space of random variables" and if it is practical to even define these sets.
So, let $X: \{\text{events}\} \to \mathbb{R} $ be any random variable with property $P$.
Is there a name for the space of all such random variables
$\mathcal{X} = \{X | X \text{ satisfies } P\}$
for some particular properties $P$?
This is not exactly an answer. It's rather a simple explanation of why random variables are rigorously defined as functions and why, in my opinion, it is worth going into measure theory.
Random variables are functions. There's a reason why random variables are treated as functions. The idea is that there is some underlying mechanism that generates the observations you get. That's a funny thing: probability theory - the theory of randomness - assumes that there is some hidden order which is unattainable by the observer.
The idea is that a random variable is a function, $X:\Omega \to E$, from a sample space $\Omega$ to an observation space $E$, but we typically don't know which function it is [plus some additional regularity assumptions I will not go into].
More so, in practice we only have access to some observations $X(\omega_1)$, $X(\omega_2)$ and so on. We don't know $\omega_i$ either. Think of it as a black box containing $X$, behind it there's $\Omega$ and some unknown mechanism feeds $\omega$'s to $X$ to generate our observations...
The sample space $\Omega$ is usually an abstract entity. It's structure is dictated by what is called a $\sigma$-algebra. For now let's think of $\Omega$ as the space behind a curtain that separates is from the observable world.
Definition of probability. We cannot know $X$ and how it generates observations. Nevertheless, we can use $X$ to probe into $\Omega$ and "measure" it.
Let me say at this point that the basis of probability theory is measure theory. Measure theory is a branch of mathematics that studies how we can measure sets. For instance, it makes sense to say that $\{1,2,3\}$ has 3 elements, $[0,1]$ has a measure $1$ and $[10, 12]$ has measure (length) 2. It also makes sense to say that the box $[1,3]\times [4,6]$ has measure (area) 4 and so on. Of course there are different ways to measure how large a set is, but they should make sense (see: definition of measure).
In probability theory, the idea is that although we cannot study $X$ in and of itself, we have can "measure" it. For example, consider a random variable $X$ that takes from some space $\Omega$ into $\{0,1\}$. Define the set
$$ \{\omega \in \Omega : X(\omega) = 1\}. $$
The measure of this set of the probability that $X$ is equal to $1$. Given that there are different ways to measure how large this set is, there exist multiple probability measures.
That said, a probability measure is a measure on $\Omega$ (allows us to measure how large some subsets of $\Omega$ are) that
Here is a (not rigorous) definition of probability. Let $\mathrm{P}$ be a function that maps (some) subsets of $\Omega$ to a number in $[0,1]$. It is called a probability (or probability measure) if
Example. Take for example the random variable that gives us the outcome of a coin toss. Then $\Omega$ is some (abstract) space and $X:\Omega \to \{H, T\}$ is our random variable. We know that "the probability of tails is 0.5 (50%)". This means that the probability measure of $\{\omega \in \Omega : X(\omega) = \text{Tails}\} = 0.5$.
Properties of P. There are lots of properties (see here for a list).
The space of all random variables from $\Omega$ to $E$. Yes, this has a name - it is the set of measurable functions. It takes some time to grasp the concept of a measurable function and you first have to understand what a sigma-algebra is. A "space of (all) random variables" makes sense only once we have equipped $\Omega$ and $E$ with a sigma-algebra (like motion makes sense only from the perspective of a reference frame). The definition of a measurable function is independent of the choice of the probability measure.
I would also recommend that you read this question on stats.stackexchange.