I never really worry about it all that much. Plenty of cats around here. Our bears are mostly black bears so unless I'm strapped with meat (hauling a carcass out), they're nothing to worry about. I'll shout, "HEY BEAR!" every hundred yards or so for a few miles if I'm hauling meat out in bear country.
Last fall, I spent 4 days in a remote coastal canyon. On the last night, I heard the most god-awful screaming probably not much more than a hundred yards from my camp. It sounded like a muted Tasmanian devil. I turned on some music on my phone & got a small fire going as a precaution. I slept fine that night. I found out after I got back that it was probably a female cat in heat.
If someday I die at the hands of a wild animal, I figure it's a fitting end. But I'll sure make him work for it.
If my kids are with me, I'll sometimes carry a firearm into the woods with us. But keeping them close avoids most potential problems with wildlife.
I also figure that it's the same as sharks while surfing. If you can see it, it's obviously not hunting you. If you can't see it, then there's nothing you can really do about it anyway so just relax & have fun. We all gotta go sometime and I'd rather be eaten by a cat than die in a hospital (though neither one really appeals to me).
Death will catch up with us all eventually. If my time comes, I'll at least go down fighting (or blissfully unaware -- either will work). Until then, there's no sense in getting too worried about things outside my control.