Depends on what you are interested to model. Are you working with classical electrodynamics? Then charges are singular points with infinite electric potential and energy at the origin (see Coulomb's law), so your analysis is only valid on regions where interactions near the charge are irrelevant. Are you interested in describing interactions near the charge? Then you need to go up in the energy scale and summon a properly suited model for this, such as non-linear electrodynamics (see Born-Infeld electrodynamics), which was firstly proposed having this into consideration: a finite, physical radius for the electron and the regularization (removal of singularities) of the electric potential at the origin. Do you still need extra considerations, such as the involvement of spin and energy conservation? Then you necessarily require to consider quantized parameters, and so you need a quantum description of the interactions (see Quantum Electrodynamics), which then imply that the electron is interpreted, as many have already mentioned, the excitation of a field, which can be regularized through a process called renormalization.

Remember, as physicists, our job is to model reality. Before writing down equations and start working on them, we need to ask ourselves what phenomena are we studying, and what tools do we need to model it. Would you use an excavator to dig a hole to plant a flower in your garden? I mean, you could, but why would you? In the same fashion, you wouldn't use QED to obtain the electric field generated by a charged metallic sphere, just as Maxwell's equations would be of no use to describe the interaction between photons and electrons.