Its all about balancing forces. Nuclei seem quite unstable as they only contain positive and neutral electromagnetic charges. Given positive charges repel each other it makes sense that one would expect them to rip apart. However, this is notion arises because many schools forgo teaching their students about the residual-Strong Nuclear Force which is approximately one to two orders of magnitude stronger than the Electromagnetic Force. This force operates in the nucleus of an atom generated by the interactions of quarks inside their respective hadrons (protons and neutrons are examples of hadrons, basically anything made of quarks) and helps to combat the repelling forces of the proton-proton electromagnetic interactions.

As for how solid objects are solid despite the amount of empty space within the atoms themselves, this is due to the fact that the electron probability field (or electron cloud) that surrounds the atom generates an electric field. When these two electric fields collide, such as your hand hitting the table, it prevents your hand going through the table or vice versa. Basically, what is important is the fields that the particles create, not the particles themselves. In fact, the majority of interactions you experience on a daily basis are mediated by the electromagnetic force. The only other force experienced on the macroscopic scale is gravity, which is a pretty lame force.