I'm trying to learn about what the state of technology is at presently. It seems like we clearly can go from nucleotide to digitally stored sequence, but can we transcribe something from the digital sequence?
Oligonucleotide synthesis is a service offered by many providers such as IDT.
To add to @Maximilian Press' answer.
There is quite a lot of research on how to store arbitrary digital data in DNA. Nick Goldman describes his research on the topic in his TEDx talk. The last progress in the field I have noticed was DNA fountain, a tranformation of bites in DNA with maximal efficiency of data storage in DNA (information / nt) and reliable recovery when sequenced afterwards.