Old English sciete (West Saxon), scete (Mercian) "cloth, covering, towel, shroud," from Proto-Germanic *skautjon-, from *skauta- "project" (cf. Old Norse skaut, Gothic skauts "seam, hem of a garment;" Dutch schoot; German Schoß "bosom, lap"), from PIE root *skeud- "to shoot, chase, throw" (see shoot (v.)).
Sense of "piece of paper" first recorded c.1500; that of "any broad, flat surface" (of metal, open water, etc.) is from 1590s. Of falling rain from 1690s. Meaning "a newspaper" is first recorded 1749. Sheet lightning is attested from 1794; sheet music is from 1857. Between the sheets "in bed" (usually with sexual overtones) is attested from 1590s; to be white as a sheet is from 1751. The first element in sheet-anchor (late 15c.) appears to be a different word, of unknown origin.