So, I can't answer about solving big problems, but I did genetic engineering research in grad school on bacteria. One could very easily conduct serious genetic engineering in one's bedroom for less that $500 or so. Of course this is fairly basic stuff, but still, you'd be amazed what is possible with very little equipment.

For example, there is a yearly competition called iGEM, which is synthetic biology competition for undergrads. Some of the stuff they do with limited resources is quite impressive.

http://igem.org/Main_Page