Breaking into a field can be a challenge without experience or other qualifications to speak of. The truth is that a lot of organizations 'require' degrees in job postings and so on; however, I personally know a lot of people in the IT industry, including myself, that do not have or are working on completing their undergraduate degrees.
There are a number of paths to reach a destination, but often times, we are only interested in the shortest or that of least resistance. In your case, being that you are interested in becoming a DBA, I'd say that though it may be beneficial for you to obtain a two or four-year degree in IT/IS, it isn't a must. I would recommend that whatever you decide to major in, find the time to take some IT/IS related courses, such as database systems development and management information systems; an introduction to local area networks will also help you better understand networks and enable you to converse with network administrators and so on, who you will work with often.
In addition, though a DMBS is a DBMS is a DBMS, you will want to figure out what platform you want to focus on -- Oracle or Microsoft for example. Each has its pros, cons and certification tracks. I'm afraid that I can't speak all that fluently on behalf of Oracle, so you should refer to www.oracle.com for more information.
Microsoft, on the other hand, has a range of certifications for SQL Server, including MCDBA, MCTS, and MCITP. You can review the full Microsoft certification catalog at <a href="http://learning.microsoft.com/Manager/Catalog.aspx?clang=en-US&dtype=Catalog&Sort=PublicationDateDescending&page=1&cats=%7bf0929131-ef93-4005-b7b7-a6d4bd2a60d0%7d">http://learning.microsoft.com/Manager/Catalog.aspx?clang=en-US&dtype=Catalog&Sort=PublicationDateDescending&page=1&cats=%7bf0929131-ef93-4005-b7b7-a6d4bd2a60d0%7d</a>.
Good luck and remember to have fun!