In Quantum Field Theory, as I understood, Quantum Fields are observable valued fields. In that sense, if $M$ is spacetime a Quantum Field is a function $\varphi : M \to \mathcal{L}(\mathcal{E})$ from spacetime into the set of operators on some Hilbert space $\mathcal{E}$, like the ones we use in Quantum Mechanics to describe states of systems. Furthermore, $\varphi(x)$ is an observable for each $x\in M$, so that it is a Hermitian operator.

The one thing I still can't grasp is: what is the space $\mathcal{E}$ on which $\varphi(x)$ acts for each $x\in M$? In Quantum Mechanics, to describe a system we pick a certain $\mathcal{E}$, and describe the system with kets belonging to $\mathcal{E}$.

Here the system is the field, but we are describing the system itself as one operator valued function. It is not clear the space where it acts.

So what is the space on which Quantum Fields acts and furthermore, how this relates to the state space of kets from Quantum Mechanics?