Fields in classical mechanics are observables. For example, I can measure the value of the electric field at some (x,t).

In quantum field theory, the classical field is promoted to an operator-valued function of space-time. But no one talks about eigenvectors of the quantum field! If I try to measure the field at one point in spacetime, I should get a real value which should be an eigenvalue of the quantum field, right? I guess the eigenvectors of the quantum field also live in Fock space?