The dimensions of an atom are of the order of an $\mathring{A}$ $(10^{-10} \ m)$. Thus,there must be large electric fields between the protons and electrons. Why,then,is the electrostatic field inside a conductor zero?

(N/A) In any neutral atom,the number of electrons and protons are equal,and they are bound together to form a stable,electrically neutral system.
Electrostatic fields in a macroscopic conductor arise due to the presence of net excess charges. According to the properties of conductors in electrostatic equilibrium,any excess charge resides only on the outer surface of the conductor.
Since there is no net excess charge inside the bulk of an isolated conductor,the internal electrostatic field is zero. The microscopic fields between protons and electrons within individual atoms cancel out on a macroscopic scale,resulting in no net macroscopic electric field inside the conductor.