First off, I'd want to point out that most to all the complicated mathematical difficulties that go into "in practice" quantum field theory are really more to do with trying to deal with fields that are interacting with each other, and thus don't really have so much to do with how and what I believe you're asking about, which is to get to know the conceptual core of what a quantum field is. And that is, fortunately, much simpler.
A quantum field is the quantum-mechanical version of a classical field, which is a system in which we assign some quantity - in fundamental physics this might be electric and magnetic field vectors, but in more higher-up applications could be, say, the sound-wave field within a solid medium, the value representing the relative compression of the material - to each point in space. That is, a classical field is just a function
$$\phi(P)$$
of a point $P$ in space. In terms of coordinates, in three-dimensional space, we'd write this as a function of three arguments:
$$\phi(x, y, z)$$
if we're using Cartesian coordinates for each point, so that $P = (x, y, z)$, for example. The value returned by this function is the value of the field quantity at that particular point, e.g. an electric field of 5 V/m (ignoring the vectorial stuff for simplicity), or an increase in pressure of 5 Pa (again, ignoring more technical complexity).
So what happens in quantum mechanics? Well, in quantum mechanics, just as when we develop a quantum theory of a moving particle, i.e. one with position and velocity, we must convert this quantity into a quantum operator: we don't know which one yet, but we first just declare it. Now the return value $\phi(P)$ no longer has type "real", but has some sort of operator type, and thus it gets a hat:
$$\hat{\phi}(x, y, z)$$
and in effect we have a field of operators, one at each point. Each operator then should operate on some quantum vector $|\psi\rangle$, representing an agent's knowledge about the entire field, so that it is possible to derive from it a field-value wave function
$$[\psi(x, y, z)](\phi)$$
which gives a probability distribution describing what is known about the field value $\phi$ at the point $(x, y, z)$. (Note this is a "curried function"; I like these because they cast it into a form that makes it more clean what is going on - we're getting a probability density function (pdf) over field values $\phi$ specific to the point $(x, y, z)$.).
So that's part 1: $\hat{\phi}(x, y, z)$ here is what constitute "observables" for the quantum field. But this doesn't really give us much insight now into the next part of the question, which is how do we build the Hilbert space part. I first want to make a note, though: while we are going to do that, technically it really is only the operators above that count and everything can be done in terms of them, the Hilbert space is simply mathematical fluff to make things easier to work with. So you could say "we are done here", but we can do it nonetheless.
To see how to do it, you should note that, in a way, you can consider the field to be as though it were an ordinary multi particle quantum system with an uncountable collection of separate "particles" (these are NOT the usual particles like electrons, photons, etc. but something more mathematical), each one corresponding to a different space-time point $(x, y, z)$ and whose "position" is the field value $\phi(x, y, z)$. Hence, just as, say, for 2 particles, you have a two-particle wave function
$$\psi(\mathbf{r}_1, \mathbf{r}_2)$$
with the two position components, here you have a $\psi$ that takes uncountably many position components, all indexed by coordinates $(x, y, z)$ or, in effect, a wave function that takes in a function which is a classical field configuration $\phi(x, y, z)$. Such a "wave function" is thus also called a wave functional, and written
$$\psi[\phi]$$
for that quantum field. Then the action of the field operator $\hat{\phi}(x, y, z)$ upon such a $\psi$ is given by
$$[\hat{\phi}(x, y, z)\psi][\phi] = \phi(x, y, z) \cdot \psi[\phi]$$
just as
$$[\hat{\mathbf{r}}_1 \psi](\mathbf{r}_1, \mathbf{r}_2) = \mathbf{r}_1 \cdot \psi(\mathbf{r}_1, \mathbf{r}_2)$$
. Hence the Hilbert space is suitably-defined equivalence classes of these wave functionals (the whole "same up to a set of measure zero" stuff), and states are the attendant rays.
In short: a quantum field represents a field quantity that is quantum-mechanically fuzzy.
