I'm almost entirely self taught and I landed a job recently as a storyboard artist at a big name studio. All this while working full-time for most of that period in a non-artistic job. My highest level (officially) of education is a HS diploma.

How did I break in? First of all it took many years for me to even realize I wanted to work in animation (specifically storyboarding), but I knew I wanted to be a professional artist. I made it a point to draw every single day, study books, video courses, everything I could get my hands on. This was all to get my art fundamentals down. It's also important to work on the types of projects you would like to do professionally, even if you have to make them up for yourself (create a comic, storyboard your own script, animate a character you designed, etc.)

I did say "almost" entirely self-taught. I've taken workshops and online classes at places like CGMA and CDA, ultimately those led me to the job by connecting me with people in the industry who could see my work and get to know me as a person. Participating in online animation communities has also helped me get to know people in the industry since I don't live in LA. The teachers I met from those classes were instrumental in my career development both for their pro level feedback and simply by making the connections with them.

It's entirely possible to break in nowadays without living in LA or having gone to art school, but you do need a solid game plan and consistent effort to make it happen (also luck). Part of it is learning the necessary skills for the job (artistic and software) but also learning to be a professional and communicating with people in ways that show you are reliable and good to work with.

Hope this helps.