If you can drive a car safely you should be able to carry and operate a gun safely too. My father was up into his 90s when I requested that his doctor have his driver's license taken away (because I and others were afraid he was going to kill somebody); at that point him carrying a gun would not have been a good idea either.
I'm guessing that an elderly person might be more likely to be the subject of a criminal assault, since criminals prefer easy victims. One of the closest times I ever came to having to defend myself was after I had a torn rotator cuff repaired and my arm was in a sling with a big pillow under it, altogether making me look like an easy victim. This knockout game liberal punks like to play where they come up behind white people and sucker punch them is usually perpetrated on old people. And when you read news accounts of self defense cases, a significant portion of them involve the elderly. So I'd say by all means carry while you're able, and always be aware of your surroundings. And don't forget that home invasions do occur, so carry even when you're in your home.
The assisted care facility my father was in had a "no guns" sign on the door, and I'm assuming that's typical. So a person in such a facility might not have the option of being armed.
If you have reduced hand strength or arthritis you might need a gun that has low recoil. I just recently got a Glock 42 (.380) for my recoil-hating wife and she loves it; it's about as easy a pistol to shoot as I've found (not counting a .22 target pistol). The little .380s like the Ruger LCP have a sharp recoil and even I don't like shooting them (and I think shooting a .44 is really fun). But the Glock .380 has mild and easily manageable recoil.
Criminal assaults often come as a surprise so a gunfight is sometimes preceded by a hand to hand fight. So some knowledge of basic empty-handed self defense should go along with having a gun. Also, the older you are the more natural you look carrying a cane. So it might be worthwhile to carry one with a little weight to it and learn some simple self defense moves with it that might give you a few seconds and a few feet to get your gun out and into position.
