So you did (what, all of one or two posts ago).
And here's my definition:
DNA System: The complete cellular system that processes the genetic instructions and data stored in DNA.
That would be the cell, then? You can't process the instructions without enzymes, you can't make enzymes without making the amino-acid building blocks, you need the cell membrane to keep everything in, etc.. There are very few parts of a bacterial cell you can lose without losing the functions of transcription and replication.
All this system does is try to replicate itself. It exists to make an exact copy of itself. The genetic instructions and code's role is to produce an exact, working duplicate.