I've been doing a combination of yoga/pilates for years which has been great for staying healthy and flexible, with a focus on core-strengthening. Of course there are other types of sports and fitness-style activities I've done and do, beyond that.
It's worthwhile to be careful and know your limits. As we all age, our bodies aren't quite as indestructible as they used to be. The better we take care of our bodies now, the less problems we'll experience as we continue aging.
My parents are in their sixties. They've both been physically fit all their lives. It shows in their energy level, emotional perspectives and mental health in general. They exude good health and as a by-product, look great!