They were full-time soldiers, period. I think it's Herodotus who has a story about the other Greeks demanding a share in command during the Persian Wars (not home right now so can't check) and the Spartans told all the Greeks present (who were sitting down) to stand up by profession (farmers, smiths, etc...) and eventually all the other Greeks stood up except the Spartans. Apocryphal or not, it gets the idea across. As to what kept them busy? Probably training/exercising to maintain their physique and police actions to keep the helots suppressed and terrorized.

This class were Spartan citizens and a very small minority of the population in Sparta/Laconia/Messenia. Below them would have been the non-Spartiates perioikoi who were non-citizens but treated better. At the very base and supporting the Spartan state/army were the helots, abused and terrorized. Somewhere in this were Spartan citizens who could no longer afford the mess fees required as a full-time Spartiate, they were a grey area, Spartans without any civic rights and probably lacking respect.