In the Feynman Lectures Vol 1, Chapter 28 (at the end of section "28–1 Electromagnetism"), it is mentioned:
For those purists who know more (the professors who happen to be reading this), we should add that when we say that (28.3) is a complete expression of the knowledge of electrodynamics, we are not being entirely accurate. There was a problem that was not quite solved at the end of the 19th century. When we try to calculate the field from all the charges including the charge itself that we want the field to act on, we get into trouble trying to find the distance, for example, of a charge from itself, and dividing something by that distance, which is zero. The problem of how to handle the part of this field which is generated by the very charge on which we want the field to act is not yet solved today. So we leave it there; we do not have a complete solution to that puzzle yet, and so we shall avoid the puzzle for as long as we can.
As a curious novice I am interested to know, at a very introductory level, as to how a modern "Unified Theory" could possibly explain such a phenomenon as quoted above, so as to gain insight of how physicists work towards explaining things?