Persistent data storage is the basis of all modern information systems. The long-term value and volume of data are growing at an accelerating rate and pushing extant storage systems to their limits. DNA offers exciting potential as a storage medium, but no practical scheme has been proposed to date that can scale beyond narrow-band write rates. Here, we demonstrate a combinatorial DNA data encoding scheme capable of megabits per second write speeds. The system relies on rapid, combinatorial assembly of multiple smaller DNA parts that are dispensed through inkjet printing. To demonstrate this approach, we wrote approximately 25 kB of information into DNA using our system and read the information back out with commercially available nanopore sequencing. Moreover, we demonstrate the ability to replicate and selectively access the information while it is in DNA, opening up the possibility of more sophisticated DNA computation.