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 electronstatic field inside a conductor zero?

Explanation

The electric field between protons and electrons binds them and their equal numbers makes atom a neutral system. Electric field can be generated when there is some excess unbalanced charge, but metals always drive excess charge towards the outer surface and therefore there is always zero net charge inside metals. Excess charge on the outer surface of a conductor generates electric field for the points outside it, and the electric field inside the volume of the conductor remains zero.