slang for "drunk," 1736, sometimes said to be from notion of a ship heavy-laden and so low in the water that small waves (half seas) wash over the deck. This suits the sense, but the phrase is not recorded in this alleged literal sense. Half seas over "halfway across the sea" is recorded from 1550s, however, and it was given a figurative extension to "halfway through a matter" by 1690s. What drunkenness is halfway to is not clear.
Old English sæ "sheet of water, sea, lake, pool," from Proto-Germanic *saiwaz (cf. Old Saxon seo, Old Frisian se, Middle Dutch see, Swedish sjö), of unknown origin, outside connections "wholly doubtful" [Buck]. Meaning "large quantity" (of anything) is from c.1200. Meaning "dark area of the moon's surface" is attested from 1660s (see mare (n.2)).
Germanic languages also use the general Indo-European word (represented by English mere (n.)), but have no firm distinction between "sea" and "lake," either by size, by inland or open, or by salt vs. fresh. This may reflect the Baltic geography where the languages are thought to have originated. The two words are used more or less interchangeably in Germanic, and exist in opposite senses (e.g. Gothic saiws "lake," marei "sea;" but Dutch zee "sea," meer "lake"). Cf. also Old Norse sær "sea," but Danish sø, usually "lake" but "sea" in phrases. German See is "sea" (fem.) or "lake" (masc.). The single Old English word sæ glosses Latin mare, aequor, pontus, pelagus, and marmor.
Phrase sea change "transformation" is attested from 1610, first in Shakespeare ("The Tempest," I.ii). Sea anemone is from 1742; sea legs is from 1712; sea level from 1806; sea urchin from 1590s. At sea in the figurative sense of "perplexed" is attested from 1768, from literal sense of "out of sight of land" (c.1300).
[1736+; fr the notion that one is like a ship low in the water and burdened so that relatively low waves, half seas, sweep over its deck]