What is the minimum distance between two opposite point charges In an overly simplistic model if I have a single negative charge, and a single positive charge they will be attracted.  I expect they will fly together.  Click.  Good luck getting them apart.  The force required to pull them apart could be impossibly huge since the distance between them is really small.  The work required to get them to any large distance must be really high.
If I wanted to write a simple point charge simulator what would be a realistic minimum distance?
Is it true that outer electrons in metals "park" in a place that is relatively far away from the protons so the force to remove them isn't too great.  How far away is it?
Just wondering...
 A: There's no minimum distance. 
Yet, as the two particles get closer to each other, they will either scatter off each other (in the quantum mechanical sense of interacting via Feynman diagrams) or form a bound system - if we're talking electron-positron (which is as close to point charges as it gets), they might become positronium, but that won't last long, if they are electron-proton, they might become hydrogen, etc.
Aside from this, at the level where quantum effects kick in, the "distance" of the charges is rather ill-defined. Even in bound states, the orbitals describing the position of the electrons around the nucleus do not have constant distance to it.
It is not the case that the electromagnetic force becomes "invalid" at any range. It is just that, on very short distances, our classical picture of the force simply providing a pull/push according to Coulomb's law is invalid - the classical force arises from the tree-level Feynman diagram (one photon exchanged) between charges, while the underlying interaction in the quantum field theory is much more complicated.
A: What we know is that the electrostatic force does not work nearby the nucleus. The interaction between the positive and the negative charged particles stops at some distance, where Quantum Mechanics describes it. These distances for small orbits are well studied. To calculate them is much more difficult, for the hydrogen atom you have the formula, for all other atoms  QM  calculations use numerical methods.
Your thought about the "fly together" of protons and electrons does happens in neutron stars under the condition of much larger gravitation.
The radius of a hydrogen atom is approx.31±5 pm. The radius of the proton is about ten thousand times smaller (smaller 1 fm).
