In New York, the main driver of high cost of living in the huge demand for space within a short distance of Midtown and Lower Manhattan. The further you get from that area (in time via mass transit), the cheaper space gets, and commercial rents are lower too, which makes for cheaper goods and services, probably no different form many other major metro areas, and sometimes cheaper due to the bigger market.
Living in Manhattan, other than, say, Inwood, is choosing to purchase a luxury product at a steep price. Likewise living in a 1 bedroom apartment as opposed to a studio.
Much of Queens is cheaper than parts of Brooklyn, indeed. Whether Queens is a good option depends on whether you're likely to work in Midtown or Downtown; Queens is better for Midtown, Brooklyn (and Hoboken and Jersey City) downtown.