There are few occupations that are as open, where the barrier to entry is as low as it is in web design and development. If you have the ability, and are willing to learn, you can get started without a formal education or having to pay for college courses. Most of what you need to know can be found free of charge online.
I remember being at OpenTech in 2010 and listening to one of the speakers talk about her work with young people. One of the teenagers she worked with had ambitions to be a TV presenter.
"Well," she said, "Let's see your show reel."
"Haven't got one, miss." came the truculent reply.
She then patiently explained to the young man that he had a camera phone and a YouTube account - that was all he needed to get started.
There are some jobs where you undoubtedly need day-to-day practice in a professional environment. But for the majority of jobs, the web now seems like the perfect school for anyone motivated enough to take control of their own learning.
(I'm aware of the inherent privilege problems here - not everyone has the time to study, or the resources to access the web or get professional equipment.)
Whether you want to learn guitar or how to be a lawyer, there are resources out there to help you get started. It's true that you will need to test your mettle at some point - and possibly pay for a professional qualification - but there's nothing to stop you getting started.
In the online / digital space it's even easier. As Rachel points out above - very few of us have formal qualifications in the areas in which we work. Building a portfolio is perhaps the most important thing you can do - in any field of work.
If you're a teenager - get a GitHub account, start blogging, participate in the professional usergroups, save a record of everything you do, engage in personal SEO. When people like me come to interview you, the first thing we'll do is search for you online - have something ready to show us.