My step-dad is 52 and still plays soccer with the family (we're a big soccer family; most of us played and/or are currently coaching).
Sure, he's not nearly as fast and he can't do it nearly as long as I probably could (29) but he's still out there. My Mum also plays in an adult league (and she'd probably be mad at me if I said her age).
While a lot of the "high impact" sports (football, soccer, hockey, etc.) are likely out of reach once you really get up there (60s? I guess it will vary person to person but mid to late 60's is likely the upper limit to really play), there are plenty of lifetime sports you can still play. Tennis is considered a lifetime sport (although I have a hard time seeing someone who's older play but I suppose if you get two people of equal age, you can play). Golf is another lifetime sport.
Hell, in terms of just plain games, you can video game until you're put six feet under. Depending on how your health holds up, you can always play a turn-based game where you don't need any twitch speed or anything like that. As long as your mind is sharp and you are physically capable of holding onto a controller (or hell, just a stylus and play on a 3DS propped up), you can game forever.
Double hell...we're also assuming games remain the way they are (sitting down with a controller). I imagine eventually, we'll just be plugging into The Matrix and then Grandpa tippy2k2 and play all the video games he wants even after his body has fallen apart due to all the cupcakes he keeps eating...
Never stop gaming.