'approachableness' originates from English compounding: from the verb 'approach' + the adjective-forming suffix '-able' (meaning 'capable of being') and the noun-forming suffix '-ness' (meaning 'state or quality').

'approach' entered English from Old French 'aprochier' (to come near), itself from a- (from Latin 'ad-') + French 'proche' (near), from Latin 'prope'. The adjective 'approachable' was formed by adding '-able' to 'approach', and later '-ness' produced 'approachableness'.

Initially it meant 'the state of being able to be approached' (literal/physical sense); over time it also acquired the broader social sense of 'friendliness' or 'openness' used to describe people or manner.