Let me see if I can be more complete than the replies above.
If you have one rifle & one rifle only, it will be a compromise. There is no such thing as a great long-range rifle that is also a great brush gun and great mountain rifle. As such, you need to find a gun that works well, if not perfectly, in most places you will be hunting.
For a deer/black bear "do all" cartridge with long-range capability, I recommend you go with one of these cartridges: .270 WSM, 7mm Rem mag, 7mm WSM, .300 Win mag, or .300 WSM. Any of these will do what you want to do & none are too powerful to be shootable without a brake. (Even with these, some shooters want a brake.) Ammunition for each is readily available at a reasonable price. If the rifle you choose is available with a WSM chambering, I would choose it but the others are still very good. Each is capable of the 800 yard shot if YOU and your rifle are. Each of these will also leave massive holes if you shoot a deer at close range, but as I said, a one-gun arsenal is a compromise.
I do not recommend a RUM for most hunters, and definitely not for a 1-gun arsenal. It's simply too much gun for many hunting situations, too expensive to shoot, too much damage to animals within 300 yards, too much recoil or too loud if you use a muzzle brake and no fun to practice with. I like a RUM, and am even looking for a .338 RUM, but this is a specialty rifle for long-range hunting and nothing else.
As for rifles, assuming you are going to shoot a stock rifle like most hunters, I would strongly recommend you look at the Savage rifles. They are generally seen as the most accurate production rifles today. Tikka rifles are also known for their accuracy. Both the Savage and Tikka are among the most reasonably priced rifles. Savage offers a closable muzzle brake that is a great idea on a magnum in that this lets you use the brake for practice & turn it off for hunting so you don't have to hunt with ear plugs in.
If you go with a Remington I recommend you immediately replace the trigger. Remington is being sued over trigger defects that have some of their rifles firing when they should not. Some Remington (and Winchester) rifles shot well out of the box but many do not. Remington (and Winchester) rifles can almost always be made to shoot well, but that means dollars to a good gunsmith, how many dollars depends on the rifle & how well you want it to shoot.
Take a strong look at the 300. wsm
