There is no notion of quantization of charge in classical electrodynamics. Charge is a continuous, infinitely divisible quantity there, and there's nothing at all that would indicate what carries the charge. The electron (or any other particle, for that matter) is not predicted by classical electrodynamics, and thus none of the classical notions of self-energy apply to it.
The quantization of charge and the electron as an indivisible particle can only be understood by quantum electrodynamics, where the four-potential of electrodynamics and matter fields are properly treated as quantum fields. There is such a thing as self energy in this picture, and it is the cause for a difference between the "bare" and "dressed" charges of the electron, of which only the latter is measurable (as the ordinary charge we know), so we should be cautious to assign any meaning to the former.1
There is nothing "holding particles together", as far as we know. At least, our best (quantum field) theories do not say anything about internal structure of fundamental particles - these things just are, and no experiment has hitherto indicated any kind of substructure of the electron.
If you think the charges "inside" an electron should repel each other and thus rip it apart, then you are commiting a category error - you apply the classical notion of force on a scale where quantum effects are dominant, and the classical thinking therefore invalid.
1"Bare" and "dressed" quantities are terms appearing when renormalizing a quantum field theory. If we define our theory through a Lagrangian, then the charge $e_0$ appearing in the matter-gauge field coupling part $e_0\bar\psi\gamma^\mu A_\mu\psi$ is called the bare charge, which has, in the course of renormalisation, to be taken infinite in some sense to yield a finite number for the renormalized dressed charge $e(\Lambda)$ which is dependent on the energy scale $\Lambda$ of the process we are looking at. That is, the measured "charge" of an electron is actually not constant when we go to higher process energies, but its coupling to the electromagnetic field becomes stronger with increasing energy, giving rise to a problem known as the Landau pole, where the coupling blows up and becomes first non-perturbative, and then infinite. Note that one often looks at the fine-structure constant instead of the charge itself, but it is essentially the charge squared.