I went to a comprehensive state school.
Outside of school, I was in swimming club, did some musical stuff (though violin lessons were mostly in school,) I did DofE, I did Young Farmers, which included Christmas dramas and public speaking contests, as well as more agricultural stuff. I was in the museum club. Friends did various things outside of school, like Guides, Girls Brigade, dance, drama, various musical things, various sports (judo, gymnastics, athletics club, netball, tennis, riding - the boys had the cricket and rugby clubs, which we supported the funds of at the bar in 6th form...) It was a small town, but loads going on if you wanted and could afford club subs.
Early in 6th form, the head took aside those of us they thought capable of Oxbridge and said if we wanted to go for it, they'd give us extra tutoring and support. (I didn't get in, but I tried.)
However, I've a close friend who works in a top public school, and the money they can throw at sports facilities and coaching is unreal. At least 4 boys left with professional sports contracts last year. They do well academically because they're selective, too.
I think you can offer a lot of extracurricular stuff outside of school, but it depends where you are and what's available, plus how much taxiing around you're willing to do. I lived out of toen with no public transport, so there was a lot of our teenage years where my parents and my best friend's parents were coordinating which of them was taking us to swimming (Mondays), YFC (Thursdays), DofE (Wednesdays)...