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 zero?

Hint: Use Gauss' law.

The protons and electrons are bound in an atom with a distinct and independent existence and neutral in charge. Electrostatic fields are caused by the presence of excess charges. But there can be no excess charge on the inner surface of an isolated conductor. So, the electrostatic field inside a conductor is zero despite the fact that the dimensions of an atom are of the order of an Angstrom.