Middle English rēchen, from Old English ræcan, reccan "to reach out, stretch or extend outward, hold forth, extend in continuity or scope," also "to succeed in touching, succeed in striking;" also "to address, speak to," also "to offer, present, give, grant."
This is proposed to be from Proto-West Germanic *raikejanan "stretch out the hand" (source also of Old Frisian reka "to give, pay," Middle Dutch reken, reiken, Old High German reihhen, reichen "give, reach out, get," Dutch reiken, German reichen "to reach, to pass, to hand, to give; to be sufficient"), from Proto-Germanic *raikijanau, which is probably from PIE root *reig- "to stretch, stretch out, be stretched; be stiff."
Sometimes 16c. spelled retch. As "to hand (someone something), give" from c. 1300. The meaning "arrive at, succeed in getting to" is early 14c.; that of "succeed in influencing" is from 1660s. Related: Reached; reaching. Shakespeare uses the now-obsolete past tense form raught (Old English ræhte).
Colloquial reach-me-down "ready-made" (of clothes) is recorded from 1862, from notion of being on the rack in a finished state.
"continuous stretch or course," 1520s, from reach (v.); earliest use is of stretches of water. Meaning "extent of reaching" is from 1540s; that of "act of reaching" is from 1560s; that of "limit or scope of extension" is from 1570s. To be out of (one's) reach "unattainable" is by 1690s.
Ah, but a man's reach should exceed his grasp,
Or what's a heaven for?
[Browning, "Andrea del Sarto"]