The Roman Republic was notable for producing an extremely large number of great men. Per capita, I know of no other civilization that came close, except for the Greek city-states which preceded Roman society and served as its template. I believe this was because the entire society was set up to produce elites who were extremely powerful and extremely ambitious.
The ambition component is relatively straightforward. The Roman ideology emphasized civic accomplishment and glory as routes to social standing, power, and immortality. I don’t know all the mechanics of how they accomplished this so thoroughly, but there are many pieces that point strongly in this direction, e.g. Polybius’s description of Roman funerals, or the practice of building public works with one’s own money to support one’s election to the Senate.
The power component was more complex. It depended on two pieces: giving elites a solid base of power, and giving elites an excellent training ground. As in most pre-industrial societies, the primary economic unit was the household (whereas today the primary economic unit is the company). Even compared to other ancient societies, Roman law and culture gave the head of household extreme power over their family. For example, children did not have separate property, including unmarried adult children, and a patriarch faced no legal punishment for killing his own children or slaves. Practices of this type, combined with the feelings people naturally have for their immediate family, made households internally coordinated to a ludicrous degree. For instance, you didn’t have to worry about your second-in-command leaving to work for a competitor, because law and custom were on your side if you physically drag him back. Thus, a skilled patriarch would have a power base that was effectively immune to most attacks short of murder. This greatly lessened the Problem of Local Focus, making the household much more formidable. The coordinated household could expand by adult adoption, or by acquiring slaves. This sometimes included very skilled slaves such as doctors or businessmen, who could be rewarded with freedom and become loyal clients if they served well.
In addition to this power base, the elites also needed the skill to wield it effectively. The Roman solution to this was simply to throw them at the world face-first. Well-connected teenagers would often serve an apprenticeship as a staff officer to a relative who held a military command. At the age when modern people are in college or grad school, Roman elites would be magistrates, colonial administrators, or military officers. A politician would have to organize and execute an election campaign while in his early twenties. And of course, a man had to lead his household as soon as he married or his father died. (Julius Caesar inherited his father’s household—and his political enemies—at the age of 16.) While the specifics varied from case to case, a Roman elite was immersed in the work of power from a young age.
Children’s education was arranged by the family rather than the state, so a father might teach his son what he’d learned in the Senate, or hire a Greek philosopher to instruct him in rhetoric, or have him direct the slaves on the farm. This allowed family traditions of knowledge to be created and passed down.
This is a high-variance strategy, and I assume some elites didn’t do very well; presumably their performance was noticed and they lost the relevant elections, or were passed over for the most important posts. However, a notable fraction learned extremely well from contact with the object of study, and these were the people who made Rome into a paramount society that could serve as the foundation for Western Civilization.