C17: perhaps from hare (in obsolete sense: harass) + scare, variant of stare1; compare helter-skelter
Word Origin and History for harum-scarum
1670s (adv.), probably a compound of obsolete hare (v.) "harry" + scare (v.), with 'um as a reduced form of them. As an adjective from 1751; as a noun from 1784.