From Star Trek: Theurgy Wiki
Although Starfleet crewmembers staffed the vital positions on board the Federation starships, civilian crewmembers were allowed to hold important jobs in the ship's science and medical departments, as well as support areas like Below Decks or the Arboretum.
Spending years on end on a starship, there were some needs that aren't going to be met by Starfleet officers. Besides lounges, starships had gyms, and if Operations didn't maintain those gyms and equipment, civilians did. So long as the ship was large enough to carry civilians, then they could do pretty much anything onboard, from working in off-duty lounges, to classrooms, possibly even some small maintenance duties.
Specialists: As part of the civilian crew, the Theurgy had people that were specialized in specific fields concerning the ship's missions before the ship had to flee Starfleet. These specialists were always graduated professionals that have earned themselves a reputation in and even outside of the Federation. (Example: Narik Cinsaj was tasked to analyse and retro-engineer the Reaver in the Fighter Assault Bay before she was killed in Episode 03, Part 3).