Approximately 8000 BC, the civilization of that world, advanced beyond even 23rd century Federation capabilities, was cast into ruin by the return of a glacial age. At that time, a vast complex was constructed for the women, while the men remained on the surface. Within the complex, everything the women could need or want was provided, all administered by the Controller, an advanced computer that they revered. The women themselves existed at a low intellectual level. Doctor McCoy theorized that disuse of their intellectual faculties had led to atrophy.
When necessary, the leader of the Eymorgs could employ the Great Teacher, a mechanism that could temporarily impart the vast knowledge of her ancestors to her, so that she might take actions on behalf of her people. Use of the Teacher was strictly limited, which implies that the ancient architects of this system had some reason for keeping the Eymorgs intellectually stunted. One permitted use was when the Controller reached the end of its usable lifespan, an event that occurred in 2268.
With the help of the Teacher and an ancient space vessel, the then leader of the Eymorgs, Kara, visited the Enterprise and used advanced surgical techniques to remove Spock's brain. Her goal was to install the brain as the new Controller. She claimed the brain would function as the new Controller for 10,000 years, suggesting that that was the first time a replacement had been required.
The Eymorgs, despite their intellectual atrophy, did have something that approximated a government – a small council consisting of the leader and several advisers. When the Enterprise crew visited the complex in search of Spock's brain, they were greeted by this body. Its leader, Kara, offered the landing party a chance to leave, but as they did not wish to leave without Spock, they refused. Presumably she extended this offer because they were not Morg.
Although all of the Eymorgs were female, they did bring their male counterparts, the Morg, into their complex on occasion. Such captives were fitted with a disciplinary mechanism that could inflict enormous pain; as a result, the Morgs quickly learned obedience. Morgs referred to the Eymorgs as "the givers of pain and delight". (TOS: "Spock's Brain")