The old world used them to record large amounts of data, most of the data have been corrupted, but the Research Center might be able to get something useful out of it.
The player can find the Data Disc in Abandoned Ruins by mining. They cannot be tracked through the relic scanner, but they can sometimes be bundled together with other relics.
They can be traded to the Church Store in exchange for unique seeds and planters or you can give them to Petra from Portia's Research Center to exchange them for new blueprints. They can also be used to obtain Research Notes after all blueprints have been obtained. Lastly, they are used in some missions like Mission: Saving the Tree Farm. The Loudspeaker and Boombox blueprints are obtained at the same time if a total of 8 Data Discs are given to Petra, if only one is chosen the price is as shown on the table, but the other blueprint cannot be obtained later. Also, the Water Storage, Water Engine, and Water Wheel are all obtained at the same time (during the Saving the Tree Farm mission) for a single cost of 6 Data Discs.