Stoic physical training was focused both on testing students' beliefs and building their mental endurance. Musonius Rufus did not care if his Stoics were under ten percent body-fat or how much they could deadlift. He was concerned that when they came face to face with pain they might choose comfort over virtue. The hard work of Stoicism involves desiring only what is good and avoiding only what is bad. Pain, according to Stoicism, is not actually a bad thing, it's simply indifferent. That's an easy enough idea to pay lip service to, but when pain stands between us and virtue, will we go through that pain or avoid it? Better to test ourselves in a controlled setting. Musonius said it this way, "the philosopher’s body also must be well prepared for work because often virtues use it as a necessary tool for the activities of life...We will train both soul and body when we accustom ourselves to cold, heat, thirst, hunger, scarcity of food, hardness of bed, abstaining from pleasures, and enduring pains."
So what exercises did the ancients use to become better Stoics? We don't really know. History has taken that from us. The glimpses that we do have fall into the category of 'voluntary discomfort.' For instance, Epictetus advised that a thirsty person could wet their mouth, but then spit out the water. Seneca would eat bland but nutritious foods for long stretches. It would be interesting to see exactly how the ancient Stoics exercised, but there's no secret sauce, we simply need to train ourselves to follow Stoicism eagerly and rigorously. It isn't difficult to devise voluntary discomforts; hard beds, cold showers, and fasting come to mind. I happen to use an ice based practice that I learned in a birthing class. The point is not to make ourselves uncomfortable for discomfort's sake. We are meant to uncover the ingrained mental habits that go against Stoic thought, experience through disciplined exercise that those thoughts are wrong, and learn to consistently choose the wiser course.
Again, here are Musonius' thoughts on the matter,