Of course you can.
Why don't you start with a home DVD, like the Lorraine Kelly or Davina one? (pick one by a 20-something by all means, I just haven't tried them, and I know my 50+ friend uses the LK one regularly) You don't need any fancy kit, just some trainers so you don't knacker your knees and back which is the same advice for any age.
Try your local council gym, most do drop in or taster classes if you don't want a membership straight away. I say council gyms because they will offer a wide range of classes and will be inclusive. Every class I attend has all ages, even the advanced ones, and the gym itself. Not saying commercial gyms like Virgin Active aren't, but there a couple of gyms here which are very much for young bodybuilder types, although they would say open to all. So visit lots.
Mainly, don't give up. The hardest thing with exercise is starting. If e.g. yoga isn't for you, try pilates. But honestly, don't discount the more active cardio classes or something with weights. Even go to the weights gym, you might find you love it. The right instructor will not turn you away.