For classical field theories with probabilistic initial conditions the classical field observables are an idealization. Their arbitrarily precise values poorly reflect the characteristic uncertainty in the presence of substantial fluctuations. We propose to employ observables based on fluctuating fields. In terms of these "statistical observables" the probabilistic classical field theory becomes a quantum field theory. Non-commuting operators are associated to observables. The quantum rules follow from the laws for classical probabilities. The ``quantum'' is part of the ``classical''. A regularized functional integral guarantees the unitarity of the quantum field theory. We discuss in detail the classical relativistic Klein-Gordon equation with interactions.