Origins
The word 'stable' in English is actually two words wearing the same disguise. The adjective — meaning firm, steady, not likely to change — descends from Latin 'stabilis,' meaning 'able to stand, steadfast.' The noun — meaning a building where horses are kept — descends from Latin 'stabulum,' meaning 'a standing place, an enclosure for animals.' Both words derive from the Latin verb 'stāre' (to stand), which in turn comes from Proto-Indo-European *steh₂-, one of the most prolific roots in all of Indo-European.
The convergence happened in Old French. Latin 'stabilis' became Old French 'estable' (adjective, meaning firm), and Latin 'stabulum' became Old French 'estable' (noun, meaning a stall or enclosure). English borrowed both forms in the thirteenth century, and the initial 'e-' was eventually dropped, leaving the identical form 'stable' for two etymologically distinct but related words.
The root *steh₂- (to stand) has generated an extraordinary family of English words through Latin alone. 'State' comes from 'status,' the past participle of 'stāre' — a state is how things stand. 'Station' is from 'statiō,' a standing place. 'Statue' is from 'statua,' something that stands. 'Stage' comes through Old French from Vulgar Latin *staticum, a standing place, a platform. 'Establish' is from 'stabilīre,' to make firm. 'Constant' is from 'constāre' (to stand together, to stand firm). 'Substance' is from 'substāre' (to stand under). 'Distance' is from 'distāre' (to stand apart). 'Circumstance' is from 'circumstāre' (to stand around). 'Instant' is from 'instāre' (to stand upon, to be present). 'Obstacle' is from 'obstāre' (to stand in the way).
Proto-Indo-European Roots
Through the Germanic branch, the same Proto-Indo-European root produced Old English 'standan' (modern English 'stand'), 'stede' (modern English 'stead,' as in 'homestead' and 'instead'), and 'stall.' German 'stehen' (to stand), 'Stadt' (city — literally 'standing place'), and 'Stall' (stable) are cognates. The breadth of this single root is a sign of how fundamental the concept of standing — of being upright, fixed, present — is to human thought and language.
The noun 'stable' has its own long history. In Roman usage, a 'stabulum' could be anything from a modest animal pen to an elaborate facility for housing racing horses and chariots. The word also came to mean an inn or lodging house — a place where travellers and their animals could stand for the night. This sense survives in the traditional Christmas narrative, where Jesus is born in a stable — a 'stabulum' — because the inn had no room.
In modern English, 'stable' as a noun has extended metaphorically. A 'stable' of writers, athletes, or racehorses means a group managed by the same person or organization — borrowed from the racing world, where a trainer's horses are collectively called a stable. The adjective, meanwhile, has become essential vocabulary in science (stable isotopes, stable equilibrium), computing (stable release), medicine (the patient is stable), and everyday speech.