Can neutrons touch other neutrons and can protons touch other protons?

When you say "touch," you are most likely seeing solid balls with a specific diameter, coming closer together than this diameter. Much like ball bearing stacked together.

Problem is, when analyzing protons in a nucleus, concepts like "diameter" and "distance between protons" become meaningless concepts. We can NOT say with any meaning what the "diameter" of a proton is -- we can only discuss the probability that one particle, at a specific distance from a proton, will interact with that proton. Worse, we also can NOT discuss with any meaning "where" a proton is located within a nucleus. At these distances, to say, "Proton 1 is here, and proton 2 is a distance of .5 fermis away," is a nonsensical statement. When we get within a nucleus, ordinary concepts of location and distance disappear.