"piece of a barrel," 1750, back-formation from staves (late 14c.), plural of staff, with the usual change of medial -f- to -v- (compare leaves/leaf). The plural form possibly was in Old English but not recorded there.
1540s, "to fit with staves," from stave (n.). The meaning "break into staves" is from 1590s (with in from 1748, chiefly nautical, on notion of bashing in the staves of a cask). Past tense stove. Stave off (1620s), however, is literally "keep off with a staff," as of one beset by wolves or dogs. Related: Staved; staving.