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Hi Op,
First off That is not an(ir)rational fear. I have woken up to grizzly bears right at my tent. I do not carry a gun, but I do carry bear spray and a marine flare gun. Learn how to indentify if it is a brown or black bear. Each encounter depends on many factors. First, what type of bear, is it a mom with cubs, or is it a sub-adult black bear walking straight towards you?
When I go into the wilds, I do not keep food near my tent, instead I use a bear bag, and tie it up in a tree, at least 100 yards from my camp site. I do not cook in the same clothes thaT I sleep in. I never keep snacks in my tent. I also have an electric bear fence, but I rarely take it with me because it is too bulky, and a pain to set up. You could always bring some fishing line, and set it up so it rings a bell if something walks into it during the night. Oh, I also wear bear bells on my hiking boots to alert them when walking along the thick overgrown trails.
Here is a pict from the bears that came around last time I was in AK.
Good luck OP!