outside
外 centers on the concept of outside: the exterior, beyond a boundary, or something other than the inside.
excluding; except (for); apart from; other than; besides; in addition to
outside; exterior
outside ...; beyond ...; not included in ...
foreign; abroad; overseas
foreign country
outside (e.g. of a building); exterior
unexpected; surprising
to be disconnected; to get out of place; to be off; to be out (e.g. of gear)
diplomacy
to remove; to take off; to detach; to unfasten; to undo
going out; outing; leaving (one's home, office, etc.)
surgery (branch of medicine)
exception
exterior; outside; outer; external; lateral
outward appearance; looks
unexpectedly; surprisingly
outside the country; abroad
outdoors; outside; open air
outward appearance; exterior appearance; outward show; looks
suburb; residential area on the outskirt of a city; commuter belt
exception; exclusion
foreign; imported
foreigner (esp. one of European ancestry); gaijin; whitey
eating out; dining out
外 combines 夕 (evening) and 卜 (divination). The traditional explanation is that divination was performed in the evening, outside the normal daily activities, giving the sense of 'outside'.
At evening (夕), a fortune-teller (卜) goes outside to read the stars. The scene of stepping out at night captures the idea of outside.
For ガイ, imagine a guy stepping outside in the evening to do some fortune-telling: guy -> ガイ.