If you used the right bullets, got close enough, and picked your shot carefully a 6.5x55SE will kill a moose just fine. At least that's what about century of Scandanavian hunting experience would tell me. As far as the "right" bullet, probably any premium 150-160 grain bullet will work well. As for range, well I don't know enough about the ballistics to comment when it comes to shooting moose. I would imagine though that would not be all that hard to figure out as it will be a combination of your skills at placing the bullets in the kill zone and the energy/velocity retained at various ranges. So, if a 6.5 Swede will kill a moose, I'm sure that with proper bullet selection and within appropriate ranges anything bigger, such as .308, will also kill them just as dead.
You seem to be asking if a .308 will kill a variety of animals. I'm assuming that's because you already have one, and want to know if it'll be effective or if you need to step up to a more powerful weapon. Aside from perhaps buffalo, there's not much NA game that can't be killed humanely with a good shot at an appropriate range from a .308. Even buffalo, if you can get close enough for a brain shot are well within the purvue of the .308.