Their primary responsibility in most city-states was the care of the household. Jobs like cleaning, childcare, nursing sick family members and slaves, making cloth, and especially cooking* all fell to the women of the household, slave and free. The woman of the household, usually the wife, supervised everyone else, taking the highest status tasks and assigning others as necessary. Babies’ wet-nurses were usually slaves, women from Sparta being particularly sought after for their ability to raise healthy children.
One household task, however, that was almost always performed by a slave woman and not by freeborn women was that of fetching water. Until the Roman period, most Greek cities did not have the advantages of aqueducts and the like, meaning that all water had to be fetched and carried from a city well or other source and respectable freeborn women did not leave the house.
The helot** women of Sparta deserve special consideration. Since Spartiate women spent more of their time on athletics and motherhood, a greater portion of the housework fell on the shoulders of the helot women. They also made nearly all of the cloth produced in Sparta, taking over the primary task performed by free women elsewhere in Greece.
As in Rome, some slave women worked more publically as entertainers (flute-players, tumblers, actresses). Prostitution was an extremely lucrative business and a significant number of prostitutes were slaves.
Freedwomen in Greece still owed service to their former masters after their manumission. They were less likely to have specialized training than their Roman counterparts and were more likely to continue in the same sort of jobs they had held as slaves, usually domestic service or textile work. There are, however, records freedwomen in other occupations such as wet-nurses and even market vendors.
Even so, it is difficult to find information on slave and freedwomen in Greece. This is in part because much of their work was basically the same as that of freeborn women and its value not quite equivalent. Neither ancient sources nor modern research has much to say about them.
*Which was decidedly not something men did, in Athens at least.
**The helots were a subject population in Sparta. They were, essentially, publically-owned slaves and were the primary farmers in the area, the Spartiates being more concerned with warfare. It is not really possible or profitable to say whether they held a higher or lower status than slaves from elsewhere.
"Occupations of freedwomen, Athens, 4th cent. BC" (nos. 329-332) in Lefkowitz, Mary R. and Maureen B. Fant, trans. Women’s Life in Greece and Rome: a Source Book in Translation. Baltimore, MD: Johns Hopkins University Press, 1992.
Pomeroy, Sarah B. Goddesses, Whores, Wives, and Slaves. New York: Schocken, 1995.