I tend to disagree with LSUMarkb, but I'm a bit bias. I've been a sys admin for 3 years or so and I don't even have a high school diploma. I spent a lot of time training myself while working in tech support and getting many certifications. It's about finding the right company to work for that shares your values and values your skills.
That being said, a BS in something hands-on and useful certainly doesn't hurt!
As far as getting your start I can only tell you what I did. I started out in tech support at a small company which also had a growing IT department. I made it clear I was interested in systems administration and engineering. I helped out with various web development projects, making tools to help people. As I learned PHP and Linux command line, I was assigned various clean-up tasks on some of the systems. After a good hard year of this continuing I was offered a position as a Jr. Linux Systems Administrator, which is when I started to realize how little I actually knew. That's when it started getting really fun. :)