Edited Repost from another thread:

Spartan society was a very stratified social hierarchy. At the top was the Homoioi, those men who had gone through the agoge and earned their citizenship and could therefore elect the Ephors and Gerousia. Homoioi translates as "those who are alike", indicating that, other than the 2 kings, Spartan citizens were among equals. The Homoioi were a military aristocracy who, as you say, spent a good deal of their time training as warriors.

Below them sat the Perioeci, "Those who dwell around", who were freemen but not citizens. They formed the the craftsman class, and could own land and fight as hoplites, but could not vote. Perioeci were the skilled workers...you might think of them as the Spartan middle class.

Spartan women enjoyed quite a bit more freedom than most other Greek women. They could own and inherit land, for instance. They could not vote, but the women generally ran the estates and handled the money, since the men were devoted to military pursuits, and therefore could wield some level of power (Aristotle condemns the Spartans for the influence of their women in Politics, 2.1269b).

At the bottom of the pyramid were the Helots. These were slaves/serfs who worked the lands owned (conquered) by the Spartans, and enabled the Homoioi to practice their dedicated military lifestyle. They considerably outnumbered the Spartan Homoioi, and fear of the Helots revolting was a major factor in Spartan foreign and domestic policy. The Ephors would annually declare war on the Helots so they could be killed more or less at will, as part of a terror campaign to keep them subdued.

The simple answer is the one by /u/RonPossible: the perioikoi and the helots did the work. However, that hardly made Sparta unique. You're wondering how Spartan society could function if a minority of men spent none of their time working - but you could ask them same about any other Greek society.

Membership of the class of Spartiatai or Homoioi was limited by a property requirement, which effectively meant that only the very wealthy could remain part of it. Defined by this property requirement, the Homoioi were not a "military aristocracy" but a leisure class - the relatively small top layer of society that was rich enough not to have to spend any time working to provide for themselves or their families.

Such leisure classes existed everywhere in Greece. It was the great ideal of all Greeks to be rich enough not to have to work. Freedom from toil would allow them to spend their time on philosophy, music, hunting, horse riding, athletic training, politics, travelling and so on - basically, living the good life.

At the top level, then, there is little structural difference between Spartan society and the society of many other Greek states. All such states featured a minority of wealthy men who did not do any of the labour that was required to allow their society to function. All relied on a more or less free majority of people from lower social strata to do the actual work.

The crucial difference is that the Spartans uniquely forbade their own leisure class from engaging in nearly all of the activities I just mentioned, and required them to dedicate themselves to a life of war. They did this to secure internal stability (removing causes for internal rivalry within the elite by outlawing means of obtaining ostentatious quantities of wealth and political power) while at the same time providing themselves with a more effective militia than those of other states.

Tl:dr; it didn't matter that leisure-class Spartans were warriors, since their equivalents in other states didn't do any work either.

slaves. the Spartans were a relatively small group who had enslaved another (much larger) group called the Helots. as a result, the spartans were constantly terrified of a slave revolt. Spartans spent their time training for (among other things) a slave uprising, and spent the rest of their time making sure the slaves knew they were slaves. one of the "tests of manhood" for a spartan male was to sneak out and kill one of these Helots without being caught.

spartans were pretty serious about making sure the enslaved STAYED enslaved.