Jeff and Koen covered the subject pretty well, but I'll toss in my $0.02:

I sum up the whole process in three words: Read, Learn, Do.

Read whatever you can. Blogs, forums, books. Books On Line (or BOL) is included with Sql Server Developer and is also available on the Web. Read up on database design fundamentals. Things like entities, relationships, normal forms, data types, etc. (Joe Celko has a very good Stairway on the subject right here on SSC). Read up on the branch of Databases that really floats your boat. If you like tinkering around with code and writing procedures or whatever, then development might be your career path. If you are fascinated (as I am) with the data itself and manipulating it to find out what it is telling you, then business intelligence and/or data science might interest you.

Learn. Always be curious and willing to invest time in learning subjects that will help. You don't necessarily need a math or computer science degree to design, develop and administer databases. But on the other hand, there are subject areas which you should at least be conversant with depending on the database speciality you want to pursue. Statistical knowledge definitely comes in handy for business intelligence and data science if for no other reason than to be able to communicate with and understand the math geeks who do have formal degrees.

Do. Like others have said, buy a copy of SQL Server Developer, volunteer your time to charities or non-profits, whatever you can to actually build the "street smarts" of databases. I started out by building stuff in MS Access. I'd build queries in the design grid then flip them over to the SQL view to see exactly what was going on. Then I'd start changing this or that to see what would happen. That led to reading up on design fundamentals (normal forms in particular), SQL syntax and so on. I found I had a knack for it. Later on I got into desktop support but now I've managed to leverage that into a position where I still do support but also have my own SQL Server sandbox (2005 Developer) on my desktop at work where I can go to town.

In the IT world in general, it's more about what you can do than what you know. Yes, there's some minimum level of knowledge, but I don't think you'll have any trouble with that. Once your foot's in the door - that's probably the hardest part - prove yourself and always be willing to learn.