Manhattan (and parts of Brooklyn) != NY. I wouldn't call it "cheap", but there are plenty of places in NYC where one can rent a bedroom for under $1,000 a month easily (and maybe closer to $500/m in some places). That, coupled with not having a car, actually makes quality of life very good if you're someone who seeks out city living.
I know plenty of people that make $30k-$40k/y in NYC and can afford to live in Bushwick with a few roommates and live nice lives. Single train commute into Manhattan, plenty of things to do that don't cost a lot of money, no need to worry about a car, etc.
Edit: a lot of people are replying that having roommates isn't tenable for $x or long-term. I agree, and I don't think we disagree.
Solving a housing affordability problem for people making $15/h starting their careers is part of the overall picture - housing available for people at every income level. An experienced worker with a family won't be looking to live in a 3br apartment with two roommates, but that same 3br apartment that was affordable can also house a family of 4 with two wage-earning parents.
NYC is _filled_ with people in all sorts of jobs who, because there is housing stock available in many levels, can afford to live alone, with a family in an apartment in Queens, or in a house in NJ.
It's not panacea - transit is not as reliable as I wish it was, goods are expensive, but it works for a lot of people