c. 1300, "fixed payment (usually in exchange for taxes collected, etc.), fixed rent," from Old French ferme "a rent, lease" (13c.), from Medieval Latin firma "fixed payment," from Latin firmare "to fix, settle, confirm, strengthen," from firmus "strong; stable," figuratively "constant, trusty" (from suffixed form of PIE root *dher- "to hold firmly, support").
Sense of "tract of leased land" is first recorded early 14c.; that of "cultivated land" (leased or not) is 1520s. A word of confused history, but there is agreement that "the purely agricultural sense is comparatively modern" [Century Dictionary]. There is a set of Old English words that appear to be related in sound and sense; if these, too, are from Latin it would be a very early borrowing. Some books strenuously defend a theory that the Anglo-Saxon words are original (perhaps related to feorh "life").
Phrase buy the farm "die in battle," is from at least World War II, perhaps a cynical reference to the draftee's dream of getting out of the war and going home, in many cases to a peaceful farmstead. The simple term buy it as slang for "suffer a mishap," especially "to die" is attested by 1825, and seems to have been picked up in airmen's jargon. Meanwhile fetch the farm is prisoner slang from at least 1879 for "get sent to the infirmary," with reference to the better diet and lighter duties there.
mid-15c., "to rent (land)," from Anglo-French fermer, from ferme "a rent, lease" (see farm (n.)). The agricultural sense is from 1719. Original sense is retained in to farm out.