I am a DBA, specializing in MS SQL, with some Oracle as well. I got into it through a operations role – we didn’t have a dedicated DBA so I got involved in SQL admin and implementation, then when they advertised for a DBA role, I asked if I could have it.
3 years later and I’m the go-to database guy and, if I may say so myself, fairly well-regarded, technically. I did my degree in Geology, and had no formal IT training before I started the operations role, so there’s no reason you can’t do the same, if you can demonstrate some technical aptitude and ability to learn.
Oracle is harder to learn – I maintain that they keep everything deliberately awkward and inaccessible to put off “amateur” admins…