In physics a "particle" is the name of an approximation in classical mechanics in which we ignore all degrees of freedom but the center of mass motion of an extended body. Obviously, the "particle" in your question is a misnomer.
The problem you are talking about arises in classical electrodynamics with Coulomb's law. If we apply the formulas for the energy density of the electric field to a point charge (which is not the same thing as a particle!) and integrate over the entire space, then we get a divergent integral. This classical self-energy is therefor not well defined in classical electrodynamics. We have a number of options to deal with this. We can say that Coulomb's law is not valid below a certain radial scale (but then we need to find evidence for that, which fails) or we can acknowledge that classical electromagnetism can not deal with self-energies, at all. Strictly speaking, though, electrodynamics is incompatible with all forms of matter and one can not expect it to describe something that lies beyond its range of applicability. Just like matter in Newtonian mechanics is "a given" without self-consistent internal explanation, point charges in electrodynamics have to be accepted to exist and to behave in certain ways, without any further insight into their nature within the theory itself.
The solution to the riddle of stable matter comes in quantum mechanics, where matter becomes the low energy expression of quantum fields. However, even there the equivalent problem (vacuum self-energy) hasn't found a physically satisfactory explanation and we basically have to treat it, for now, as an unsolved issue.
