The electron is, as far as we can tell, a "point" particle, meaning that it has no internal degrees of freedom: all its degrees of freedom have to do with its relation with the surrounding space(-time).

The classical radius of the electron is the radius of a sphere such as the energy of the electric field outside it matches the observed rest mass of the electron. If you assume a "classical" electron is truly pointlike you get an infinite energy for its electric field.

The funny thing about quantum field theory is that, since the electron is pointlike (see first paragraph) you need to "renormalize" the self-interaction of the electron (i.e., the interaction of the electron with its own electric field) and renormalization methods involve a "cutoff" (effectively, a mass or a radius cutoff - see second paragraph). You then ger "running coupling coefficients" which means that the "bare mass" and the "bare charge" of the electron vary with the "cutoff".

We have met the enemy, and he is us — Pogo