If positive charges and negative charges attract while negative charges repel each other, then why don't the positive protons within an atom repel?

They do, it is just that the strong nuclear force, a force different from the electromagnetic force you are referencing, is much stronger and overwhelms the repulsion. It only acts when things are very close, as the internal distances of the nuclei of an atom, but it is incredibly strong. The electromagnetic force acts over vastly longer distances. While it would take a lot of energy to overcome the electromagnetic force repelling two protons to get them close enough for the strong force to fuse them into a helium ion, it can happen. Gravity, the electromagnetic force, the weak nuclear force, and the strong nuclear are the four known forces. The weak and electromagnetic forces merge at very high energies into the electroweak force, but that is getting too nuclear for me. Make sense? In a star, the forces are so huge at the centers that protons can get close enough for the nuclear force to take over and fuse into heavier nuclei.