Assuming that in the ancient world, it was just some sort of open air gymnasium, in a big plot of dirt, in the direct sun, what type of training exercises would they do, what wouldn’t they?

The first thought is I am highly dubious that an individual would lie on their back, and try to press something really really heavy, while on the floor. You wouldn’t want to get your back dirty.

It would have to do with lifting, throwing, carrying, stuff like that.