As so often happens, the answer to this question depends on what you mean by the words used.
Quantum fields are not things in the universe, but rather postulated quantities that are used in an attempt to provide descriptions of possible states of that universe. A state of the universe described in terms of such fields may or may not have mass, depending on the situation and on what you mean by “mass”.
And “mass” is also problematic. The parameter called “mass” in the equations defining a quantum field theory is related to the mass of a minimal (“single particle”) excitation of that field – and indeed this can sometimes be zero (as in the case of the Electromagnetic field for example). But a single photon does have energy and Viktor Toth tells us that Einstein said “E=mc^2” and so it must have mass. So what gives?
Well, when Einstein said “E=mc^2” (or perhaps a bit after the first time he said it) he understood that this was just in the particular situation of a system in its rest frame and a single photon does not have a rest frame!
For states involving many photons it may be possible to define a specific frame of reference by requiring some condition of balance between the observed frequencies (eg for two photons, the point from which they both appear to be departing in opposite directions with equal frequencies might perhaps be taken as the centre of mass). And in that case the force required to accelerate the entire system away from rest might be used to define a rest mass of the system that includes the kinetic energies of its components (in this case just the energies of the photons) as well as the masses of those components (which in the case of photons would be zero).
Source: (1001) Alan Cooper’s answer to Do quantum fields have mass? – Quora