"inside diameter of a gun barrel," 1580s, from Middle French calibre (by mid-16c., perhaps late 15c.), often said to be ultimately from Arabic qalib "a mold for casting." Barnhart remarks that Spanish calibre, Italian calibro "appear too late to act as intermediate forms" between the Arabic word and the French.
But English Words of Arabic Ancestry finds that the idea of an Arabic source "comes with no evidence and no background historical context to support it. It is far more likely that the word was formed in French" from Medieval Latin qua libra "of what weight" (a theory first published 19c. by Mahn), from fem. ablative of quis (from PIE root *kwo-, stem of relative and interrogative pronouns) + ablative of libra "balance" (see Libra).
In U.S., expressed in decimal parts of an inch (.44-caliber = ".44-inch caliber"). The earliest sense in English is a figurative one, "degree of merit or importance" (1560s), from French. Later, figuratively, "the capacity of one's mind, one's intellectual endowments."