I am not a musical person, in fact I am trying to learn the guitar now which sort of made me think of this question. How did the Greeks, or any ancient peoples, tune their musical instruments? From cursory research people say they either just made sure they were in relative tuning with each other or that they used a tuning fork, but if it was a tuning fork, how did they know that the fork was the correct tone?
The tuning fork has only been around since 1711 so there is no way that the ancient Greeks used a tuning fork to tune their instruments.
The tuned musical instruments that we know of from ancient Greece are primarily string and wind instruments. There are some percussion instruments, including bells, but it seems that few (none?) of the percussion instruments were tuned to definite pitches and that bells in ancient Greece were largely used for signaling and ceremonial purposes.
The string instruments were easily tuned, similar to how modern string instruments are tuned by adjusting the tension in the strings. The wind instruments could be tuned less easily which means that if a string instrument were to be played with the wind instrument, the string instrument would be tuned to the wind instrument.
I suspect, though, that the question is really about how was a pitch reference maintained and used in ancient times. And the answer is that...it wasn't! Before large-scale manufacturing and global shipping became available musical instruments (and their musicians) did not tend to travel far. Of course, the ideas of music and the instruments spread far and wide, but the actual instruments didn't tend to go far from where they were built. Thus, instruments played together in an ensemble need only be in tune with each other in order to sound good when played together. There is a reasonably good history of the pitch standard on Wikipedia, which is worth reading. So, the use of relative tuning is exactly what musicians have done throughout history, and would be a likely method for ancient Greek musicians.
A question that we might ask is: how often did these musicians need to tune their instruments to play with other instruments? It's not clear to me if we know enough about the musical traditions of ancient Greece to know how common it was for instruments to be played in an ensemble. It seems the ancient Greek musical traditions were primarily centered around instruments accompanying vocals. Given that surviving ancient Greek musical scores are still being interpreted for performances by musicians today, I'm also not sure if there is a definitive answer to this. It seems likely that at some point in ancient Greece there should have been instruments attempted to be played with each other. I suspect the musicians would try to tune the instrument that is most fixed in pitch and then play to that particular tuning.