You have to understand quantum mechanics to really see what's going on. A glib, if imprecise, response would be to blame it on the Heisenberg uncertainty principle. An electron cannot be confined to a small area without having an uncertainty in its momentum. So the smaller the box you put it in, the more energy it has. (Note: any time a physicist invokes the uncertainty principle to explain anything, keep one hand firmly on your wllet).
The precise answer is that you have to solve the schrodinger equation to find the allowed states for the electron. It's a differential equation not too unlike a classical wave equation. Just as a standing wave in a tank can only have certain discrete wavelengths, an electron can only have certain energy levels. And just as a standing wave has a minimum (fundamental) frequency, the bound electron has a minimum energy level.
So it would take a force much stronger than electrostatic attraction to bind an electron in the nucleus. What happens in electron capture isn't so much that the electron gets bound in the nucleus. It just has to find itself there, and if it is energetically favorable to do so, a proton can grab it in a weak interaction and turn into a neutron (and spit out a neutrino).
And no, electrons do not feel the strong interaction which is what binds quarks into nucleons (among other things) and binds the nucleons together in the nucleus.