Undergrad Problem with classical electrostatic potential energy

[zonde]

In classical physics electrostatic potential energy is: $U=k_e\frac{q_1q_2}{r}$
So amount of potential energy is not limited as $r\rightarrow 0$
But obviously potential energy (= binding energy) is limited by masses of charge carrying particles. Say when electron and positron annihilates their combined mass is converted into photons. So it would seem like annihilation happens at non-zero overlap of two particles.
Well, for me it seems like a problem that needs an explanation. Say, are charges not constant?
As QED is current most complete description of electromagnetic phenomena, I would like to ask if QED holds some explanation for this problem (given I correctly identify it as a problem)? May be there is ready explanation for this problem to which you can point me?

 

[Orodruin]

You do not need QED. Ordinary quantum mechanics works fine, giving a ground state with finite binding energy.

 

[hilbert2]

The potential energy of an electron-positron system can temporarily have a negative value of arbitrary magnitude, but then it isn't necessarily an energy eigenstate (with a hydrogen-like wavefunction, $1s, 2s, 2p_x$...). The electron-positron annihilation has to be dealt with by QED. If you force an electron and positron really close to each other and they annihilate, the resulting photon energies have to contain the potential energy of the initial state that had a short electron-positron distance.

 

[zonde]

Okay, I understand that this is not a problem in QM because there is ground state energy level that does not exceed maximum of binding energy (determined by sum of unbound particle rest masses). So to understand it better I would have to try to understand why the ground state has the energy it has.
Thanks Orodruin and hilbert2, now I have an idea where to look further.