i hear a lot of .308, but i would like to point out that shooting .308 out of a handgun would be miserable and unrealistic for anyone less than 6'4" and 220.
of course, we are in the rifle forum....so in that case my vote is 7.62x54r.
why? because ballistically it is more powerful than the .308; its equitable to the 30-06. BUT its about 1 cm shorter than the 30-06. generally better choice, IMO.
if, however, we are taking into account the necessity of using handguns (REAL handguns, none of those cut down AR's and AK's which get sold as handguns), i would go with .30 carbine. big enough to take deer (and people) at limited ranges and small enough to be fired out of a realistically sized revolver by most people.
EDIT: i would like to add that the OP did include law enforcement and self defense. so the cartridge has to be able to be fired in a pistol.