Idealization of Eletric Field at a point According to Jackson "Classical Electrodynamics", in the first chapter about electrostatics:

[...] point charges or electric fields at a point must be viewed as
  mathematical constructs that permit a description of the phenomena at
  the macroscopic level, but that may fail to have meaning
  microscopically.

I think I've understood why point charges represent an idealization, but why also the electric field at a point fails to have meaning microscopically?
Moreover, what does it mean in practice that in electrostatics we are only interested in macroscopic phenomena? Does this also affect electrodynamics?
 A: 
why also the electric field at a point fails to have meaning microscopically?

You have no problem with a point particle being an idealization, right? Now, remember
$$
    ∇.E = ρ
$$
If you put garbage in the right hand side, do not expect anything better out of the left hand side.

what does it mean in practice that in electrostatics we are only interested in macroscopic phenomena? 

If you knew the position of a charged particle with absolute precision (assuming for now that it is still a point), then, by the uncertainty principle, you would know very little about about its velocity. Since the velocity is basically unknown (and because moving charges produce magnetic fields), you cannot have pure electrostatics at the microscopic level where quantum mechanics prevails. In the classical regime, where your charged body is made out of many charged particles, electrostatics will appear because the tiny magnetic fields average out.
By the way, if someone tells you that 'virtual photons are not real particles', ask back 'what is particle?' They might try to handwave something about virtual being temporary. Ask them then 'what about resonances?'
A: We ususally talk about the EM field (as E and M together) as per QFT, and the electric and magnetic field are manifestations of the EM force.
Now you are asking why the EM field must be viewed as a mathematical construct. This means that the interaction of the EM field with other charges is mathematically modeled by virtual photons.
Virtual photons are not real particles, they are a mathematical method that we use to describe the phenomenon, when an EM field has a effect on the fabric of spacetime so, that in the region where the EM field exists (near field), it will have an effect on spacetime so that any particle that interacts with the EM field will have an altered trajectory.
These fail to have a meaning at the microscopic level, because in reality, we do not know how the EM field interacts, what we do know is that we do experiments, and we use virtual particles as a mathematical model to describe this phenomenon, and our data fit this model best (QFT).
