dragnet n.

late 14c., draggen, "to draw a grapnel along the bottom of a river, lake, etc., in search of something;" late 15c., "to draw away by force, pull haul," from Old Norse draga, or a dialectal variant of Old English dragan "to draw," both from Proto-Germanic *draganan "to draw, pull," perhaps from a PIE *dhregh- "to draw, drag on the ground" (source also of Sanskrit dhrajati "pulls, slides in," Russian drogi "wagon," doroga "way;"  connection to Latin trahere "to draw" is possible but problematic).

Meaning "draw (feet, tails, etc.) along slowly" is from 1580s; intransitive sense of "move heavily or slowly, hang with its weight while moving or being moved" is by 1660s. Meaning "to take a puff" (of a cigarette, etc.) is from 1914. Related: Dragged; dragging. Drag-out "violent fight" is from c. 1859. To drag (one's) feet (1946 in the figurative sense "delay deliberately") supposedly is from logging, from a lazy way to use a two-man saw.

Old English net "open textile fabric tied or woven with a mesh for catching fish, birds, or wild animals alive; network; spider web," also figuratively, "moral or mental snare or trap," from Proto-Germanic *natjo- (source also of Old Saxon net, Old Frisian nette, Old Norse, Dutch net, Swedish nät, Old High German nezzi, German Netz, Gothic nati "net"), perhaps originally "something knotted," from PIE root *ned- "to bind, tie." But Boutkan says it has no clear IE etymology and implies it might be a substrate word.

From late Old English as "light, open woven fabric used as protection from annoying insects." From late 15c. as "light, open mesh bag for the hair."