There is more than one way to become a webmaster or web developer. I do not have a degree in Computer Science or Design, I graduated in Linguistics, and went into Technical Writing. From there, I moved to writing for the web, and then took some C programming classes and became a webmaster.
First, try to get a job in a company that has or could use a website, doing something like Technical Writing or Technical Support. Once you have your foot in the door, spend as much time as you can building web pages and learning about how your company handles their website. Make friends with the web team members. Offer to do some of the more annoying jobs (like maintaining the site, fixing bad links, and so forth) in your "free" time. This is good resume fodder if nothing else. You'll also have the chance to see what it really is like to work in a web department. Find out from the manager of the group what you would need to be considered as an applicant, and then go out and do that.
A Computer Science Degree is Not Always Required
There are a lot of jobs that state that they require a Bachelor's degree in Computer Science, but if you prove you can do the work, you'll get hired even without that. web development is more than just programming, you need to be able to write, and design good looking web pages. Focus on what you're good at and not what you don't know.
Here are some specific articles that will help you get a job as a webmaster:
- Getting a Job as a webmaster
- I've Learned HTML, What's Next?
- Do You Want This Job?
- A Day in the Life of a Webmaster
I have put up a library of links on how to get a job in the web field: Jobs in Web Design / HTML / Web Development