Problem with Maxwell's theory 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? 
 A: 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.
A: One issue with classical electromagnetism is mentioned in Introduction to Electrodynamics, Third Edition by David J. Griffiths, in the section starting on p. 465 about the "radiation reaction", which Griffiths describes this way:

According to the laws of classical electromagnetism, an accelerating
  charge raddiates. This radiation carries off energy, which must come
  at the expense of the particle's kinetic energy. Under the influence
  of a given force, therefore, a charged particle accelerates less
  than a neutral one of the same mass. The radiation evidently exerts a
  force ($\textbf{F}_{rad}$) back on the charge—a recoil force, rather
  like that of a bullet on a gun.

He then goes on to show that Abraham-Lorentz formula, $\textbf{F}_{rad} = \frac{\mu_0 q^2}{6 \pi c} \dot{\textbf{a}}$, "represents the simplest form the radiation reaction force could take, consistent with conservation of energy"--he notes that he hadn't performed a true derivation but that "As we'll see in the next section, there are other reasons for believing in the Abraham-Lorentz formula." He goes on to say:

The Abraham-Lorentz formula has disturbing implications, which are not
  entirely understood nearly a century after the law was first proposed.
  For suppose a particle is subject to no external forces; then
  Newton's second law says
$F_{rad} = \frac{\mu_0 q^2}{6 \pi c} \dot{a} = ma$
from which it follows that
$a(t) = a_0 e^{t/\tau}$,
where
$\tau \equiv \frac{\mu_0 q^2}{6 \pi m c}$
(In the case of the electron, $\tau = 6 \times 10^{-24}$ s.) The
  acceleration spontaneously increases exponentially with time! This
  absurd conclusion can be avoided if we insist that $a_0 = 0$, but it
  turns out that the systematic exclusion of such runaway solutions
  has an even more unpleasant consequence: If you do apply an external
  force, the particle starts to respond before the force acts! (See
  Prob. 11.19.) This acausal preacceleration jumps the gun by only a
  short time $\tau$; nevertheless, it is (to my mind) philosophically
  repugnant that the theory should countenance it at all.

Then he adds a footnote saying:

These difficulties persist in the relativistic version of the
  Abraham-Lorentz equation, which can be derived by starting with
  Liénard's formula instead of Larmor's (see Prob. 12.70). Perhaps they
  are telling us that there can be no such thing as a point charge in
  classical electrodynamics, or maybe they presage the onset of quantum
  mechanics. For guides to the literature see Philip Pearle's chapter in
  D. Teplitz, ed., Electromagnetism: Paths to Research (New York:
  Plenum, 1982) and F. Rohrlich, Am. J. Phys. 65, 1051 (1997).

