"forepart of a ship's main deck used as a hospital," 1580s, from sick (adj.) + bay (n.2), in the sense "forepart of a ship between decks, forward of the bitts, on either side," from the notion of a recessed space.
"unwell," Old English seoc "ill, diseased, feeble, weak; corrupt; sad, troubled, deeply affected," from Proto-Germanic *seuka-, which is of uncertain origin. It is the general Germanic word (compare Old Norse sjukr, Danish syg, Old Saxon siok, Old Frisian siak, Middle Dutch siec, Dutch ziek, Old High German sioh, Gothic siuks "sick, ill"), but in German and Dutch displaced by krank "weak, slim," probably originally with a sense of "twisted, bent" (see crank (n.)).
Restricted meaning "having an inclination to vomit, affected with nausea" is from 1610s; sense of "tired or weary (of something), disgusted from satiety" is from 1590s; phrase sick and tired of is attested from 1783. Meaning "mentally twisted" in modern colloquial use is from 1955, a revival of the word in this sense from 1550s (sense of "spiritually or morally corrupt" was in Old English, which also had seocmod "infirm of mind"); sick joke is from 1958.
"opening in a wall," especially a space between two columns, late 14c. from Old French baee "opening, hole, gulf," noun use of fem. past participle of bayer "to gape, yawn," from Medieval Latin batare "gape," perhaps of imitative origin. Meaning "compartment for storage: is from 1550s. Somewhat confused with bay (n.1) "inlet of the sea," it is the bay in sick-bay and bay window (early 15c.).
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<a href="https://www.etymonline.com/word/sick-bay">Etymology of sick-bay by etymonline</a>
Harper, D. (n.d.). Etymology of sick-bay. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/sick-bay