How can the fact that atomic nuclei are so stable be explained, when they are formed by particles with charges of the same sign? Don't charges of the same sign repel each other?
Nuclear stability can be explained by considering what the  binding forces are like and what the binding energy of the nucleus itself is like.

a) Binding forces:
For the nucleus to be stable there has to be some type of attraction which gives rise to a force of attraction among the protons which can overcome the electric repulsion of their charges.
We know of the existence of the Strong Nuclear Force  (SNF) with nuclear scope, which binds the quarks to form protons and neutrons: the existence of this force explains the stability of nuclei.

b) Binding energy:
The mass of protons and neutrons when separated and when joined together to form the nucleus are not the same, and is slightly smaller when they are joined. This difference in mass in called the mass defect M D.
Where is this mass? It has been converted into the binding energy of the nuclear components.

The equivalence between mass and energy is deduced from Einsteins's equation: D E = D mc².