It depends! Greeks faced this problem and solutions were proposed and tried. At first chronology was kept by refering whos era was it, so for example we can say that Athens were founded during the "time of Theseus" etc.

But for the most of the time local magistrates were used as a dating tool. So for example if you wanted to refer to e.g 377 BC, you would say that somethin happened during the archoncy of Nausinicus. Of course almost every city used their own rulers for chronology so it wasn't universal. Athenian Archoncy was most commonly used, especially during and after the Hellenistic period, it was sometimes also used by Greeks even during the Roman times. (Arrian)

It, of course, wasn't the only way, Herodotus if he had problems with dating something he would use generations as a tool, with 3 generations consituting a century. Eratosthenes, the "father of chronology", promoted use of Olympiads as a system, as they happened regularly every four years, so you could easily pinpoint how many years are there for the next Olympiad.