"Good" equipment isn't always necessary. I'm pretty sure Solace uses a $10 microphone, and his songs sound nice enough. I've used a $100 Shure SM58 and have it come out terrible because of cables, so it's all about what you do with what you got.
As far as a private area...do you have a car? When I was 16 I recorded a handful of songs in a car, just because I could get heated in there without disturbing anyone. It sounds weird, but it actually comes out decent considering the fact that you're rapping in a fucking car. You just have to use a laptop. Shit, I guess now they have multitrack recorders on iPhones, but I can't imagine the vocals come out too great.