What is the force holding the nucleus together and what does it need to overcome?
The strong nuclear force binds protons and neutrons together.
Well, clearly it must overcome proton-proton electrostatic repulsion in the nucleus, and indeed it does. Check this wiki link here.
The strong nuclear force is apparently very short range; nevertheless it is about 100 times greater in magnitude than the electromagnetic force, which is a good thing for nuclei! Why?