Old English wudu, earlier widu "tree, trees collectively, forest, grove; the substance of which trees are made," from Proto-Germanic *widu- (source also of Old Norse viðr, Danish and Swedish ved "tree, wood," Old High German witu "wood"), from PIE *widhu- "tree, wood" (source also of Welsh gwydd "trees," Gaelic fiodh- "wood, timber," Old Irish fid "tree, wood"). Out of the woods "safe" is from 1792.
"building for storage," 1855, earlier "light, temporary shelter" (late 15c., Caxton, shadde), possibly a dialectal variant of a specialized use of shade (n.). Originally of the barest sort of shelter. Or from or influenced in sense development by Middle English shudde (shud) "a shed, hut," which survives, if at all, in dialect, from Old English OE scydd.
<a href="https://www.etymonline.com/word/woodshed">Etymology of woodshed by etymonline</a>
Harper, D. (n.d.). Etymology of woodshed. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/woodshed