Well, you don't need to be in your 60s to feel old, I hadn't hit 30 yet when I got to feeling a bit old:
I was 28 and managing a shop in a local mall, we had a job opening to fill and the company had a very firm rule that we weren't to hire anyone under 18 years old. I was weeding through the applications and seeing birthdates from the early 80s and thinking "I can't hire them, they're just kids" Then the horror of the math hit me like a brick, at the time, folks born in 80 or 81 were hitting their 20s and I COULD hire them!
Footnote: that was just a few years ago, I just hit 31 last month.