Did the ancient Greeks have zero in their number system?

During the classical and early Hellenistic period (until 200 BC) Greeks did not use any positional system, they had their own which was decimal but not positional. The units from 1 to 9 are assigned to the first nine letters of the archaic Greek alphabet from $alpha$ to $theta$. Each multiple of ten from 10 to 90 was assigned its own separate letter from the next nine letters from $iota$ to ϙ. And so on, see Greek numerals. This worked fine for practical and even some scientific calculations. In the 3rd century BC Eratosthenes and Aristarchus, Archimedes's contemporaries, did calculations for estimating the circumference of the Earth and distances to the Moon and the Sun without the benefit of not only zero but even of a positional system.

However, there weren't enough letters to express very large numbers (which is where placeholder zeros may come handy), no more than the myriad ($10^4$) could be expressed. When the need arose Archimedes in Sand Reckoner invented his own number system capable of expressing them. He introduced "orders" based on the myriad, which was the "unit of the first order". The myriad myriads ($10^8$) was unit of the second order, and so on. Archimedes did not have zero either, but one can manage without it even for something like a positional system. Archimedes's was akin to the system with base $10^8$, but it did not take.

After 200 BC Hellenistic astronomers, such as the father of astronomy Hipparchus, started using the sexagesimal system (with base $60$), imported from Babylon along with long term precise astronomical data Greeks lacked. This gave a great boost to the development of mathematical astronomy, and the sexagesimal system was sometimes used for doing astronomical calculations, in particular by Ptolemy in Almagest in the second century AD (presumably he was a Greek ethnically, but he lived and worked in Alexandria, Egypt). However, the sexagesimal system made no advances beyond a very narrow circle of practitioners, not even all astronomers used it. Zero was used as a punctuation mark for purely notational purposes, not as a number, and the symbol for it, when not simply blank space, was $omicron$ for "obol" (a coin of smallest value). For details and further developments see Was the concept of zero ever developed without relation to positional number systems?