how special is nuclear force?

I think your question couldn't be answered using theories in classical physics. It needs an understanding of quantum mechanics and perhaps quantum electrodynamics (?).

I am not an expert in qunatum physics. But to put the matter in simple terms, the binding force between nucleons is a "strong nuclear interaction" force, which is one of the four natural forces (the other three forces are gravitational force, electromagentic force, and weak nuclear interaction).

Strong unclear interaction is explained by the exchange of "gluons" between the quarks, which are the basic component making up protons and neutrons. Hence, proton and neutrons are NOT elementary particles. Electron, however, is itself an elementary particle. It contains no quark. Exchange of "gluons" between an electron and a proton is thus not possible. That said, it is physically impossible to "bind" an electron and a proton together. The question of "binding energy" or "mass defects" bewteen an electron and a proton thus appears meaningless.