Why do we describe physical systems in Hilbert space? In quantum mechanics we study physical systems associated with a Hilbert space. Why do we need a Hilbert space to describe the state of a system?
 A: Although there are many perspectives on this, largely united by the (correct) notion that Hilbert spaces allow for geometric tools to be applied, I'd like to present another overlooked perspective: Hilbert spaces are the tool of choice in QM because they can uniquely mimic probability spaces.
In quantum mechanics, we are given an object that entirely describes the physical system under consideration, which we'll label $\psi$, and are tasked with generating a mapping from the space of $\psi$ functions to $\mathbb{R}\in[0,1]$ assigning probabilities $p(\psi)$ to them.
This process involves defining a probability space $(\Omega,\mathcal{F}, P)$ where $\Omega$ represents all of the different observable $\psi$ (call them $\hat{\psi}$), $\mathcal{F}$ represents the different possible "combinations" of  $\hat{\psi}$'s (this is called a $\sigma$-algebra), and $P$ is the mapping that associates a probability $p$ with each $\hat{\psi}$ or "combination" of them.
Although Hilbert spaces are not probability spaces, they are uniquely up to the task to mimic these if we do the following:


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*Associate all observable $\psi$'s, the $\hat{\psi}$'s , with $\Omega$.

*Equate the inner product of a $\hat{\psi}$ with itself to the measure $P$.

*We can decompose the space into an orthonormal basis, which allow us to define any $\psi$ as the linear combination of a set of the observable states $\hat{\psi}$. This is unique to Hilbert spaces.  We can then use these to properly define $\mathcal{F}$.


The fact that we can make this decomposition is what lets us properly define a probability space, and Hilbert spaces are the only ones that can do it (at least straightforwardly) because of the notion of orthogonality they induce. (Note that $L^2$ is the only $L^p$ space that defines orthogonality, precisely because it's a Hilbert space.)
A: A short answer but I'll post nonetheless. 
Usually quantum mechanics is presented in such a way that the Hilbert space is the most fundamental object in the theory, and everything else arises from that initial structure. That is, states are elements of the Hilbert space and observables are represented by operators on that Hilbert space.
There is another approach in which observables are represented by elements of what is called a $C^*$-algebra (see note at bottom). This is the fundamental postulate in this approach. One can define states as functions which act on these $C^*$-algebra objects.
It can be proven that there is an isomorphism (Gelfand-Naimark Theorem) between these two approaches. This means you can basically derive one from the other.
I would argue that, in a physical sense, the second approach is somewhat more natural. This is because one can ask, what are the observables in my system, a very physical question. One then has to write down corresponding mathematical objects which obey the rules of a $C^*$-algebra, but then all of the business with states and probabilities can follow later.
This is in contrast to the Hilbert space approach where we begin with the Hilbert space which at first second and third glance doesn't really have an obvious connection to the physical systems we are working with and there is this sort of awkward thing where observables "act on" states which takes some getting used to. In the $C^*$-algebra approach one is able to think of observables as being represented by numbers (just like in classical mechanics) but one has to recall that these are special numbers that don't commute.
Anyways, I've given a very poor explanation here. Here are some more references if you are interested.


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*arXiv:1211.5627

*The $C^*$-algebraic formalism of quantum mechanics, J.J. Gleason, 2009.


All this to say we don't necessarily need a Hilbert space for quantum mechanics.
Note on $C^*$-algebras: The most naive way to think of a $C^*$-algebra is that its elements are pretty much just like the usual complex numbers we are used to but they don't necessarily commute upon multiplication. This probably doesn't capture everything important about them but it can get you pretty far.
