[length of cloth] Old English sciete (West Saxon), scete (Mercian) "length of cloth, covering, napkin, towel, shroud," according to Watkins from Proto-Germanic *skautjon-, with notions of "corner," from *skauta- "project" (source also of Old Norse skaut, Gothic skauts "seam, hem of a garment;" Dutch schoot; German Schoß "bosom, lap"), from PIE root *skeud- "to shoot, chase, throw." "A very abstract and uncertain semantic development" according to Boutkan (who rightly can't resist adding a German assessment of it, etwas hervorragendes).
It is attested by mid-13c. as "large square or rectangular piece of linen or cotton spread over a bed next to the sleeper." The sense of "oblong or square piece of paper," especially one suitable for writing or printing on, is recorded by c. 1500; that of "any broad, flat, relatively thin surface" (of metal, open water, etc.) is from 1590s. Of a continuous sweep of falling rain from 1690s. The meaning "a newspaper" is recorded by 1749.
Sheet lightning, caused by cloud reflection, is attested from 1794; sheet music is from 1857. Between the sheets "in bed" (usually with sexual overtones) is attested from 1590s (played upon in "Much Ado"); to be white as a sheet is from 1751. The first element in sheet-anchor (late 15c.), one used only in emergencies, appears to be a different word, of unknown origin, perhaps with some connection to shoot (v.) on the notion of being "shot out."