This is the only route I know of to get past the absurdity of "entry-level" job descriptions requiring 2–4 years experience.
Pretty much everyone in the field knows that you can train an enthusiastic and reasonably intelligent newbie up to a useful junior dev in about 2 months. What they don't know is whether you're an enthusiastic and reasonably intelligent newbie. The skills I mention above - reading your boss's mind, learning things you don't know, and convincing people to take a chance - are far more important than any technical skills you can learn.
Work on a couple of projects, alone or with friends. Learn new stuff on your own. Be confident. From my experience this works like a charm.