You said sheep and goats right not wooly mammoth, well I'd say 26" barrel 243 or 25-06 the best optics you can afford and as said 500m to point blank won't work so I'd pretend I'm in the U.S. and pack 45 auto loader maxed out for Bobo.
You have obviously never hunted either mountain sheep or goat and as I stated earlier this isn't a bear defence thread.
Most shots for our Dall or Stone sheep will be from 50 ft to 450 mtrs in an uphill or downhill scenario. Wind is ALWAYS a factor as is less than perfect shooting position. A 243 or 25-06 does not have the bullet weight or BC for this kind of hunting, a 264 WM is minimum and a touch light if you wanted to take yon grizzly over there in the saddle mooching blueberries. I personally have no use for the .277 bore so next up is a good fast 7mm with good bullet weight, 160 and up and then the bigger 30s which in my opinion are perfect.
Goats are a much bigger animal and take a lot more killing, than most people realize. Plinking them with a 243 or 25-06 just isn't the best idea around.
Our goats are not easy to hunt like some places in B.C., the country is ugly, steep and the only access is on foot with backpack for miles, then you start climbing up into the crags. After 2 or 3 days of glassing and playing peek-a-boo with them you may finally get a shot at a good billy and it may be at any angle and any distance out to 500 mtrs. You must assess his accessability if he drops on the spot and you must also try to guess where he may go if he doesn't. You must also assess where he will go if he falls and can you get there. Goat hunting in the Yukon is not for the faint of heart or those out of shape, let me tell you.
Grizzlies are the least of your concern as you are hopping from ledge to ledge, with a 3000 ft drop below you, trying to "get around back" and up and over the chosen billy. Sheep hunting is somewhat the same but in much less forbidding surroundings, and all the grizzlies I've seen while persuing these 2 animals were either oblivious to our presence or were looking for another mountain to play on in a hurry.
Any caliber suitable for this type of hunting is more than adequate for our grizzlies up here. Because of our game laws, grizzlies are in much greater danger from sheep and goat hunters than the other way around. Anyone can buy a tag for grizzly and all GMUs are open season for residents.