Yes, we have some ancient Greek records but they are not what you may think they are. The ancient Greeks living from about the year 750 BCE to the year 146 BCE used several materials to create “records”: stone, wood, parchment, and papyrus. Some ancient parchments and papyri survive to this day. The wooden writing surfaces have not been nearly so well-preserved. But thousands of Greek inscriptions written in stone have been discovered and cataloged by scientists.
Although it would be impossible to describe a public records system as organized as that of Rome (which conquered Greece in 146 BCE), the Greeks were avid readers and writers. Much of our knowledge of ancient world literature and history (and mythology) was handed down by the Greeks, whose works were preserved in libraries across Europe and the Mediterranean.
Through the discipline of epigraphy, scientists and historians assemble indexes of Greek inscriptions and cross-reference them with other written sources to compile biographies of long dead Greek citizens, to confirm accounts of deeds and events that have been called into question, and to study the evolution of Greek writing.
There are a few online archives where you can find transcriptions or images of Greek inscriptions. And there are also Websites that publish lists of other Websites where you can find this information.
In addition to formal records such as grave inscriptions we have some of the historical books that Greek writers made. But many of them have been lost to time, perhaps due in part to their parchment being reused by later writers. But it is possible we may be able to rediscover some of the lost Greek classics. In 2013 researchers announced they had found traces of ancient Greek writing on medieval manuscripts. This discovery, combined with the technology used to read the “lost” writing, gives us hope of discovering more ancient writings that may have been replaced on highly valuable parchment.
To learn more about this subject, check out the epigraphy page on the Greek Language Website.
You can also browse some more general purpose resources such as Aphrodisias in Late Antiquity, which covers both Greek and Latin inscriptions.
This is a links page on Archive.Org for many useful resources. To visit the actual Websites, you will have to modify any links to remove the Archive.Org prefix, but you should be able to see all these pages’ archives without modifying the links.
The Perseus Collection is an online archive of translations of ancient Greek texts into English.
The Eagle Portal is another important resource for people who study ancient Greek and Latin inscriptions online.