Could commercial or private schools be offering some sort of vocational degree meeting those requirements? Do you just walk in off the street at the corner Starfleet Recruiting Office?
I have no military experience at all, so I'm trying to figure how they convince civilians to enlist. Let's assume they don't always send out a ship captain to rescue you from a bar fight.
The only non-coms we really have any background on are Chief O'Brien and Simon Tarses. Chief O'Brien, as mentioned above joined Starfleet to avoid going to a music academy. Simon Tarses wanted to go into space, but didn't want to spend four years in a classroom first, so he went the enlisted route.
Really, I think you'll find similar reasons for everyone who serves in Starfleet in the enlisted ranks. For whatever reasons they aren't interested in an extra four years of education (college/university isn't for everyone) and starship command isn't one of their career goals anyway. Obviously, not very many people have this mindset given how few non-coms we've seen compared to how many officers there are.