late 13c., "quarrel, estrangement, discord, strife," from Old French destance (13c.), from Latin distantia "a standing apart," from distantem (nominative distans) "standing apart, separate, distant," present participle of distare "stand apart," from dis- "apart, off" (see dis-) + stare "to stand" (see stet).
Meaning "remoteness, space between things or places" is late 14c. The figurative sense of "aloofness" is the same as in stand-offish. Phrase go the distance (1930s) seems to be originally from the prize ring, where the word meant "scheduled length of a bout."