There is a large rigorous mathematical literature on quantum field theory. This literature is not complete, in that it does not cover the QFTs of greatest interest to particle physicists, the 4d Yang-Mills theories. But it does cover a great many other models, both toy examples and statistical models used in condensed matter theory, and it also covers the perturbative aspects of the Standard Model in complete rigor.
What follows is an idiosyncratic view of this literature.
The ur-text in the subject is Streater & Wightman's PCT, Spin-Statistics, and All That. This book doesn't have a lot of examples, but it does cover non-interacting scalar and spinor theories. It should at least answer your questions about Fock spaces.
There's also the algebraic QFT approach, which focuses on the properties of the operator algebra of local observables. The classic book here is Haag's Local Quantum Physics. I confess the algebraic QFT approach is not really to my taste, so I'm giving it short shrift here.
An interesting related read is Robert Wald's QFT in Curved Spacetime and his papers with Stefan Hollands. The language is not explicitly rigorous -- it is written more for physicists than mathematicians -- but it is careful and correct. Worth looking at to see some of the limitations of the flat space QFT axioms.
If you want to get closer to particle physics, there is a considerable literature on rigorous perturbation theory. Urs Schreiber has a nice overview of the subset of this literature that intersects with algbraic QFT. Also very much worth looking at is Kevin Costello's book Renormalization & Effective Field Theory, which cover perturbative QFT from a Wilsonian point of view.
Finally, and not least if functional analysis is what you want, there's the 'constructive quantum field theory' literature, which focuses on the rigorous mathematical construction of interacting Euclidean QFTs.
The standard text here is Glimm & Jaffe's "Quantum Physics: A Functional Integral Point of View". This book describes the Euclidean path integral rigorously and constructs interacting scalar and spinor theories in spacetime dimensions 2 & 3. Also worth looking at are Rivasseau's From Perturbative To Constructive Renormalization, and Battle's Wavelets & Renormalization.
As I alluded to, this program aimed at eventually constructing the relativistic 4d theories we use in particle physics. It has not succeeded. Partly that's because the analysis is quite difficult in 4d. Partly, it's because some of the components of the Standard Model (the $U(1)$ and Higgs fields) probably can't be made rigorous, as they seem to become trivial in the continuum limit. I think it's reasonable to hope for a construction of a non-trivial gauge theory in our lifetime, but a lot of the energy in constructive QFT has in recent years gone into studying systems that emerge in statistical physics. A nice jumping off point to this literature is the IAS/Park City lecture collection Statistical Mechanics.