I'd argue none of these individual were geniuses though. By all means they were men that made considerable contributions to society but I think they were simply men that had opportunity and means to pursue their passions.
Aristotle almost certainly never worked a day in his life. He likely benefited from patroi-cliens arrangements. His job was to think, to learn, to observe the world around him. We don't really have anything like what he and his contemporaries had today, the closest we have is career academics.
Leonardo (his name is NOT Leonardo da Vinci, that would be like calling me Ryan Speedway) was again someone who mostly lived off of patrons/benefactors after his youth, which we know virtually nothing of, and had a lot of free time. He went into an apprenticeship at 14 and by 28 he was working for, arguably, the richest family in the world as an artist. Again, he simply had a lot of free time to contemplate the world around him and tinker.
Benjamin Franklin read. A lot. Like really a LOT. By 21 he was hardcore networking by creating Junto and later as a Mason. Combine his reading with his mechanical knowledge by working with printing presses and being generally successful financially again gave him a lot of free time for the era, time to spend contemplating existence, observing the world around him, talking to experts in other fields, sitting around Lodge discussing various topics.
I wouldn't say they were geniuses, I would say they were fortunate. They mad means of support and had free time. They had the freedom to pursue things that interested them and support systems in place to allow them to worry little about surviving and focus more on tinkering.