Later, teenage years, I was also diagnosed with flat feet, needed orthotics. I don't wear them now, I probably should though as my ankles do get sore. This also lead to back problems that I went to the physio for when I was 19.
When I was 17 I tore the ligament in my left ankle playing Rugby. That one hurt for ages whenever I ran or walked until last year where I tripped over in a car park and ended up on crutches because I'd torn it again. Think I might have torn it clear off, because it no longer burns when I run or walk. So that's a win!
I recently started playing basketball again and broke my Scaphoid (left wrist) only 4 games in. It's nearly finished healing... I hope.
I don't know how serious your arthritis is, but if it's manageable then you're almost certainly better off exercising than not - your body will only deteriorate more if you don't.
That's not to say I exercise anywhere near as much as I should (just walking the dogs at the moment), however, at least personally I don't use injuries as an excuse not to exercise, if I don't it's purely because I'm being lazy.
On the plus side, since I started to put on a bit of a gut my back pain has decreased because I've improved my posture by almost always tensing my core to suck in my belly ;)
New year's resolution, more exercise!