Let's consider electric field due to a charge distribution at a point $P$ inside the charge distribution. Due to inverse square nature of electric field, electric field due to an element volume charge $(\rho \ dV)$ very close to point $P$ will be very large compared to element volume charge far from point $P$. So if we consider element volume charge infinitely close to point $P$, the electric field would be infinite. So due to superposition of electric field, the net electric field at point $P$ should be infinite. Why is it not so?
So if we consider element volume charge infinitely close to point P, the electric field would be infinite... Why is it not so?
Maybe it is so - we just cannot force them to get infinitely close to each other (whatever that means) or measure the field, when they get close enough, which probably happens when particles collide.
When particles collide, they neutralize each other (like an electron colliding with a proton and producing a neutron), bounce off each other (like two electrons) or turn into other particles (like two protons). In any of those scenarios, we cannot exactly tell what electrostatic forces they experience at the moment of collision or measure these forces, but we can tell that these forces are not infinite, because the energies involved in the collisions are not infinite.
In summary, the idea of infinite electrostatic forces at infinitely small distances is just theoretical.