In the back of Piazza San Gaetano, alongside the tower, you will find the entry point to a very interesting treat, and a very historically important place for understanding Naples as a city, especially its age. Here you will find the La Neapolis Sotterrate, or the ancient underground marketplace.
The Ancient Neapolis was founded back in the fifth century BCE by the Greeks. Many of the artifacts and architecture of this place date back to this time. The market, or Macellum, was a two story structure that ran along underneath the current structure of the convent which now stands over top of it. Many of the streets in this area start with Vico and not Via, indicating the ancient Greek origins (all streets that start with this have a corresponding ancient road structure beneath them).
The entry fee is minimal and they have guided tours available in English. We found the tour very informative and important in trying to appreciate the age of city of Naples. Here is a slide show with what you will see if you have time to consider this historical adventure.