Old English scyrte "skirt, tunic," from Proto-Germanic *skurtjon "a short garment" (source also of Old Norse skyrta, Swedish skjorta "skirt, kirtle;" Middle Dutch scorte, Dutch schort "apron;" Middle High German schurz, German Schurz "apron"), related to Old English scort, sceort "short," from PIE root *sker- (1) "to cut," on the notion of "a cut piece."
Formerly of the chief garment worn by both sexes, but in modern use long only of that for men; in reference to women's tops, reintroduced 1896. Bloody shirt, exposed as a symbol of outrage, is attested from 1580s. To give (someone) the shirt off one's back is from 1771. To lose one's shirt "suffer total financial loss" is from 1935. To keep one's shirt on "be patient" (1904) is from the notion of (not) stripping down for a fight.
word-forming element meaning "lacking, cannot be, does not," from Old English -leas, from leas "free (from), devoid (of), false, feigned," from Proto-Germanic *lausaz (cognates: Dutch -loos, German -los "-less," Old Norse lauss "loose, free, vacant, dissolute," Middle Dutch los, German los "loose, free," Gothic laus "empty, vain"), from PIE root *leu- "to loosen, divide, cut apart." Related to loose and lease.