score:8
After some digging I found this:
"AKHIBTE has taken the house of Mashqu from Mashqu, the owner, on a lease for one year. He will pay one shekel of silver, the rent of one year. On the fifth of Tammuz he takes possession. (Then follow the names of four witnesses.) Dated the fifth of Tammuz, the year of the wall of Kar-Shamash."
That's a Babylonian rental contact. It's dated to the year of the wall of Kar-Shamash, which seems to be a year under Hammurabi, so it's probably from the 18th century BC.
I can't find anything about any estimates of how common it was though. It was clearly common enough to have somewhat formalized contracts as above, with mentioning of which date you take possession etc, but that doesn't mean very much.
Upvote:3
As Lennart said - this was done as early as Rome.
http://en.wikipedia.org/wiki/Apartment#Rome
The lower floors were typically occupied by either shops or wealthy families, while the upper stories were rented out to the lower classes..
Reference: Gregory S. Aldrete: "Daily Life in the Roman City: Rome, Pompeii and Ostia", 2004, ISBN 978-0-313-33174-9, p.79f.