Not nearly as dangerous as the cars those kids were driving a year or two later.
It's deeply unfortunate that the parental attitude towards tools these days is never "learn to use it safely" but rather "Hide it away! Don't you dare let MY kid near filthy manual labor!"
The college that I went to had all the engineers learn how to run a metal-turning lathe. ME, EE, BME, I think even the Computer Engineers had to learn how to run a lathe. They put us through a safety class first, which mostly consisted of how not to run a lathe (no loose clothes, no long hair, no jewelry below the elbow, no necklaces, etc). The only injury that I know from that class was from when we learned how to run welders - one of the guys caught his shirt on fire with an oxy-acetalene torch.
Yeah, lathes are dangerous, but no more or less dangerous than a lot of other things in the world. Cars are dangerous. Turning on a light switch in the wrong context can be dangerous. If you approach this stuff with the proper respect and caution, you should be fine.
Lathes are only seen this way because they can go very big. The instant death accidents happen with the big ones, especially the ones designed to quickly remove lots of material from big metal pieces.
Otherwise lathes aren't more dangerous than any other piece of machinery with exposed moving pieces. In particular the ones designed to turn small pieces of wood can't do much harm.
Don't leave the chuck key in, wear eye protection, no loose hair, no loose clothes, don't be stupid and you will be fine.
It's deeply unfortunate that the parental attitude towards tools these days is never "learn to use it safely" but rather "Hide it away! Don't you dare let MY kid near filthy manual labor!"