The economy of classical Greek (so from 480-323 BCE) varied a lot between different city states, with Athena and Sparta being almost polar opposites. And most of our evidence comes from Athens, as a matter both of what has survived, and what has been studied by modern historians. So my answer will focus on Athens. 

Classical Athens had a market economy with private property, trade, coins, and wage labour. An estimated quarter of male citizens in Athens didn't own land and had to have other ways of earning an income. There were also the metics, freed slaves or foreign-born non-citizens, who appear to have been a substantial amount of Athens' population. There were bankers who were wealthy enough and influential enough that we know their names, such as Pasion, originally a slave. There's even some evidence of long-distances connections between grain markets across Classical Greece. 

There were of course numerous differences between the Athenian economy and our own, it was a slave society, agriculture was a far bigger share of the economy than it is now, women could not vote or own land in much of Greece, and also government involvement in the economy was much less (understandable given the much more limited communications technology available then). 

Classical Athens wasn't that unusual amongst ancient economies in having private property, wage labour and markets. There's evidence of private land ownership in Ancient Egypt, Ancient China, and in Ancient India. 

Research by economic historians into medieval England has also overturned the idea of a wide-ranging change in economic organisation, between medieval times and the Industrial Revolution. To quote the economic historian Gregory Clark:

The more we learn about medieval England, the more careful and reflective the scholarship gets, the more prosaic does medieval economic life seem. The story of the medieval economy in some ways seems to be that there is no story.

Back in the bad old days, when the scholarship was less careful, the medieval economy was mysterious and exciting. Marxists, neo-Malthusians, Chayanovians, and other exotics debated vigorously their pet theories of a pre-capitalist economic world in a wild speculative romp. But little by little, as the archives have been systematically explored, and the hypotheses subject to more rigorous examination, medieval economic historians have been retreating from their exotic Eden back to a mundane world alarmingly like our own.

There are strong, I think convincing, arguments that the term 'capitalism' is fundamentally not useful as a guide to understanding historical economies. 

Source 

Engen, Darel. “The Economy of Ancient Greece”. EH.Net Encyclopedia, edited by Robert Whaples. July 31, 2004. URL http://eh.net/encyclopedia/the-economy-of-ancient-greece/