I think that that ep was the one where they landed on a planet called "Angel One"?
Anyway, on that planet the women are physically and mentally stronger than the men (Supposedly the opposite of our world; 60's, remember) and so the men do all the housekeeping while the women do the running of the cities, etc. So it's less of a master/slave situation than a flipped wife/husband situation. It was exaggerated slightly, of course.
The S.S. Odin, another ship, crash-landed a bit before the Enterprise arrived, and the men on there didn't like the fact that the women were in charge, and got all agro. So they were banished.