Electric Field Inside matter The dimensions of an atom are of the order of an Angstrom. Thus
there must be large electric fields between the protons and electrons.
Why, then is the electrostatic field inside a conductor/matter zero?
 A: Right up close to any atomic nucleus the electric field is huge, really enormous. But at a distance of 1 nanometre or more from an isolated neutral atom, the field is almost zero because the contributions from electrons and protons cancel each other out.
In a solid material, similar statements apply, except that the distance between atoms is such that you never get right away from one atom before running into the next, so you never get to the place where the electrons of a single atom cancel the nuclear field of that atom. Now the field at any one spot is the total from all the nuclei and electrons around, and it can be quite big at one point, but it will vary in direction from one point to another, with the result that when you average over a region of size a few nanometres, the average field is very close to zero. It is this average field which people are referring to when they say there is no electric field in a conductor. For greater precision, allow the region you are averaging over to be larger still, say one micrometre diameter. Now the average field is extremely close to zero, and for a conductor there is a further consideration.
For a conductor, the situation is such that if there were a field on average, then the conduction electrons would move. So they keep moving until they take up positions throughout the conductor such as to provide a field which very precisely balances whatever average field there would otherwise be inside the material. But this balancing can only cancel the average field, not its spatial variation at the atomic scale.  
I confess I have not bothered to add quantitative statements to say what the words "huge" and "extremely close to zero" mean here. But you can estimate them yourself. Use Coulomb's law for the field at the edge of an atomic nucleus. For a region of size 1 micrometre or so in an insulating solid, estimate how far from zero the net charge inside that region could plausibly be. (If the solid is polarised, it can be non-zero). For a conductor, the feedback mechanism provided by the movement of the conduction electrons means that once steady state has been achieved the field (strictly speaking, the chemical potential gradient, but let's not get into that ...)  is zero at places where conduction electrons can move freely. Inside an atom they can't move freely. On distances that extend over many atoms, they can (but quantum theory is needed to understand this).
A: The length scale of interest for classical EM is much larger than Angstroms. We are interested in scales where matter can be approximated as a continuum. At those scales the fields of the electrons and protons cancel out quite precisely for most materials. 
A: Indeed, although nuclear electric fields are screened by electrons, the screening is not perfect. At angstrom, subatomic scale there are variations in the electric field. At nanometer scale these are already largely ironed out.
When discussing electric field strength, atomic style units should be used. For example, the normal definition of E is Newton per Coulomb. Newton per elementary charge is more reasonable, as is the Bohr radius as length unit. In such units the E field takes values of the order of unity. I am guessing and leave it to the op to check. It is also clear the a very strong macroscopic electric field has only a moderate impact at atomic scale. 
A: The most important part of this question is scale. In the macroscopic scale, a conductor that is not under the influence of voltage will appear to have no electromagnetic field**. When going into smaller scales (read atomic and quantum scales) the millions of different electromagnetic fields become distinguishable from one another.
In simpler terms, imagine you have a thousand red balls and a thousand blue balls spread randomly in a field. When you are standing in the field you can clearly see which ones are red and which blue. You can even see some zones with more reds than blue and viceversa. Ok now fly one mile up in the sky. The apparent size of the balls is smaller and it is more difficult to make apart the red from the blue. Then fly another thousand miles higher. You cannot distinguish any balls anymore. They are the same to you now.
To put you into scale, an Angstrong is 10^-10 metres. If you had balls with a diameter of one metre, to represent what you could sense with normal apparatus, imagine flying at least a million miles away from the field and look at the field.
In reality what happens is that electromagnetic fields are oriented. When orienting these fields, materials do it randomly***. Because of this randomness there is not dominant orientation of the fields and all the fields cancel each other (remember that opposite fields cancel each other).
** This is not always true. Sharp objects such as needles have an alignment of the electromagnetic fields near the tip. This makes it easier to sense this field. This is the reason a magnetized needle will point to the north.
*** Also the material you measure influences this result. For example, Neodimium has a strong alignment of its fields making it a great magnet material. Some Ni-Fe alloys show this property as well.
