I Want to Become a Database Administrator - How Can I Start?
I have a feeling that this may be off topic and/or opinion based and may get closed, but I'll answer anyway for now. This is from a Microsoft SQL Server perspective, but it really applies well across the board in my experience.  I've been a DBA or a DBA consultant for about 15 years now and I've hired and been on the hiring team for junior, mid-level and senior DBAs. Others may have differing experiences in other answers..
On Certifications
I don't look at them. Especially when I am hiring someone who is relatively new. If you want to use a certification process to properly and effectively study to learn and to find where your gaps in knowledge are - then definitely. If you think the certification will make you hirable in and of itself - it just doesn't typically hold true. But it is a catch-22, some recruiters will ask for those in job requirements. But then so many people just pass low-level certs with brain dumps and a complete lack of testing. When I see a resume with little experience and a lot of certifications, I get nervous.
I guess the takeaway on this topic is: Certifications can help, but they are seldom most important or even top of criteria
On Becoming a DBA
My first question back to someone who wants to become one is why? In fact if you interviewed and told me that you really wanted to become one I'd want to hear why. Most of the best DBAs I know fell into it from development, system administration or other career paths. This does not mean that you won't be successful if you want to be one. I just like to know why.
If it is because you have the right mindset, you like troubleshooting, you like to keep things in order, you like to make sure systems are alive, data is protected and you are drawn to the career field? Then that is great.
What Skills Are Important?
To a new DBA? I want to see you have a proficiency in the basics of being a DBA. You need to know what the most important job is (and back it up with a why and how you will do it). My personal opinion is Recovery - because without the ability to recover you don't have a DBA and how I'd do it - I'd focus on the restore end and work hard at making sure that everything we did had an eye towards a possible restore. The implementation, the testing, the best practices in place, etc.. Now is that the only right answer? I don't know. If you said security and you explained it right, I'd be interested in hearing more.
Basically, though, I want to see that you have the right character traits (below) and the right basic knowledge that you should have at your skill level and experience. Starting out? I want to see that you can backup and restore.. That you can do some basic investigation into current activity. I want to know that you understand some of the basic building blocks of performance (what resources are important and why, how DB Design and development affect things) and have a lot of the character traits I describe below in place or heading there.
Character Traits
I wrote a blog post about this (Six reasons I won't hire you) awhile back. I won't regurgitate everything I said there here- but basically a good DBA is someone who has:

Great Troubleshooting skills - if you are a scatter brain and try 65 things to fix one problem and don't even remember 2 of the things you tried - I am going to pass on you if I can root that out. You have to be a calm troubleshooter with a methodology and an ability to discover and work out problems. This isn't just DB problems, but it should be verified in life and any problems one may encounter.
willingness/desire to learn and grow - I want you to be always striving to learn the next thing. Spend some of your after work time reading books, going to user groups and community events. It is a tight market out there - I want the people who are continuing to improve themselves.
Common Sense - I am starting to think you can't teach this.. Have some before you interview.
A little touch of paranoia - you are the DBA or want to be. You are about to manage a lot of important sensitive data. I want to see you be fair and easy to get along with, not arrogant, but I want someone with a bit of paranoia. I want someone who doesn't just trust by default or without verifying. You are asking to have the keys to a database environment that is important.

So How Do you Start?
Entry level jobs. Perhaps getting a job as a developer or general IT admin with some DB skills on the job. Going to community events and user groups and learning and applying yourself. Trying some volunteer or helping gigs with non profits or start ups. Basically get near databases. Work out a career path where you can do something with data.
I started out working as a support rep. Dealing with all sorts of questions, but SQL Server questions were one of the types. I studied hard, became a go to person for tough SQL problems and moved up from there. Then I worked as a Jr. DBA someplace and the rest is history.
Patience
I was just discussing this question with a friend on twitter and they mentioned &quot;you don't get there overnight&quot; - That is great advice. I see a lot of people starting out who want to be the senior DBA right now. So you need a little patience and humility thrown in there.
The first years of a DBA career are learning, figuring out which way you want to go and a lot of support time doing tasks that are basic. While you learn in these tasks and disciplines and prove yourself, more and more tasks will be added and more responsibility granted. If you stick with it, work hard, keep things online, build the right character and grow in your skills continually, you will get to senior DBA and do more fun stuff (well combined with more meetings and time spent with project managers) - but it is a marathon and not a sprint, in the DBA world.
These answers are opinions, and that is why this whole thread not survive, but if it helps you out - copy and paste it and give some of the advice a whirl. Best of luck!