TDS: Too Damn Simple. First, think of the technology needed to make the basic steam engine. It couldn't be Hero's Engine: To do any actual work, Hero's Engine would have to use a lot of steam. As it's a reaction device, with acceleration x mass of steam going out equaling mass x acceleration of arms going around, to do any work, Hero's Engine needs to use a lot of steam, because steam is quite light and so it's necessary to make up for that by making it go faster, just like a rocket's exhaust. But the mass of the steam will to some extent be flung out as well as backwards ( like the similar lawn sprinklers, which lose a lot of their energy in spraying the water- if they were perfect turbines, the water would simply fall after leaving the nozzles), so there's going to be a lot of steam wasted, so there's a need for lot of steam. To make a lot of steam go fast, you need a boiler big enough to transfer heat to a lot of water, and that big boiler has to be under a good bit of pressure. Now, one of the reasons the Newcommen engine could actually work, in the early 18th century, was the boiler could be under very little pressure- it was very difficult , and dangerous, for them to build something that big that could be under much pressure. So, we can assume that, if it was hard to do this in 1730, it would be hard some 1,720 years earlier.

It is possible that they could have constructed a Newcommen engine- they knew something about casting brass and bronze, had blacksmiths who could forge and weld. Casting a steam cylinder would have been a difficult task, as well as scraping the inside to a decent bore, getting the iron together to make all the pieces...but maybe it could have been done. But the Newcommen engine was a happy accident; he was trying to make a Savery engine, which wouldn't work, but a flaw in the steam cylinder allowed a jet of water to condense the steam there, and the atmospheric pressure pushed the piston down. In other words, an ancient Greek would have had to first build the entire engine, with just the right flaw in the casting, to make it work.

On more problem: a steam engine in the early 18th century pumped water, typically out of mines. That's because mines ( especially the tin mines in Cornwall) had gone deep enough to become flooded out- it was impossible for any other kind of pumping machine, whether water wheel or human powered, to keep them drained. It wasn't concern for the workers- the miners were still worked to death, really. The Greeks would work their mining slaves to death, too, but don't seem to have had a problem with flooded mines, yet; they didn't know how to dig deep without having the shafts collapse, so didn't. Figuring out how to dig deep would happen much later. So that also didn't involve steam engines, and slavery didn't matter in that, either.