What exactly is the problem with classical Maxwell theory and the blowing up of energy at $r=0$? Does it have any other problems on the classical level?
Answer
There is nothing wrong with Classical Electrodynamics. Electromagnetism is an effective theory in the sense that it provides an almost exact description of physics in our everyday life energy scales, but has a few technical problems like the ones you mention. This is due to the fact that classical electromagnetism is just a very good approximation to a more deeper theory, one we now realise as Quantum Electrodynamics(QED), which takes into account the relativistic quantum mechanical description of maxwell's equations to tackle these problems.
For the above problem physicists have realised that in the proper quantum mechanical description of charged particles that interact via the electromagnetic force is through the emission and absorption of virtual gauge bosons, in the case of QED, photons. This means the separation between two particles cannot physically go to $0$ because they will interact and exchange momentum (via the gauge boson) much before they physically occupy the same spatial state. There are of course other ways this could be avoided like the exclusion principle preventing this from happening to two interacting fermions, or the discrete energy states of electrons in atoms (and thus a stable ground state at a distance $r > 0$) in the case of electrostatics.
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