20-Year Prospect
Almost any career dealing with technology shows growth promise over the next twenty years. Who knows, by then you may have your own robot to do the parts of your job you don't want to do.
Thankfully, the likelihood of robots taking over all jobs is more "bad movie plot" and less "reality." Most jobs, including database administration, require a human touch and the understanding of what it takes to do business that only the human brain can master. It's hard to imagine artificial intelligence ever matching the broad skill base needed to be a great database administrator.
It's equally as hard to fathom DBA duties being outsourced to some overseas firm. You might use cloud storage for some things, but companies need real live, breathing, thinking, and interacting DBAs to make their businesses thrive.
Technology will continue to grow and change our world to some degree over the next twenty years. Great database administrators will be needed to keep up with that technology and make information accessible for a long time.