Not sure about Alaska, but in Michael Piatt's history of the Bodie CA mines of the late 1800's, he discovered that the miners favored cheap concealable revolvers called British Bulldogs. They were considerably cheaper than the more famous Colt "Peacemaker" , or Single Action Army, and unlike the Colt they also could be carried in a pocket.

For hunting, all sorts of rifles, shotguns and rifle/shotgun combinations were available of if the miner had the money, and a small lever-action rifle in 32-20 would have been very handy for getting small game for dinner. But if you want to assume the miner was about as poor as most miners and was genuinely worried about bears, a good possibility would be a 45-70 Trapdoor Springfield. It had been made in large numbers from 1873-1885 for the US Army, was replaced for in 1892 by the Krag and handed over to militia units, and those units in turn mostly discarded it when the 1903 Militia Act created what's now the National Guard. The result was a huge number of them came on the market and for decades in almost any small general store there would be one hanging behind the counter with a price tag of anywhere from $1 to $2. The 45-70 ammo for the Trapdoor could also be easily reloaded with black powder, and around 1900 reloading ammunition for newer cartridges was more difficult, as smokeless powder was hard to buy. Another cheap surplus rifle would be the Swiss Vetterli. The Swiss had dumped thousands of this magazine-fed rifle on the market when they modernized to the Schmidt Rubin in 1890, and quite lot of them made it to the US, distributed by surplus dealer Francis Bannerman and even Sears & Roebuck. One would set you back about $7, in 1900, and they ended up in many out-of-the-way places in the US and Canada, quite a few even featured in the Battle of Blair Mountain in WV in 1921.