Welcome to Yellowstone National Park.
In a world this cold and bitter, the geothermal areas become the few places where mankind can strive to survive. The geyser basins in Yellowstone provides life sustaining heat. Gas oxygen is still present. The residents can breath without special preconditions. The plants don't froze to death, as artificial light sources keep the lesser plants growing.
The residents use gathered materials to build themselves makeshift homes. with old woods, cloths, and metal plates, they build themselves a safe haven, where life is hard, but at least there is still life.
(This place doesn't appear in the original short story.)
The world of A Pail of Air -- Earth had deviated form it original orbit, flying away from it's precious sun. Without the sun, photosynthesis stops, mass panic occur, the apocalypse had come.
Years later, the temperature drops below -200°C. Most part of the world had become nothing but darkness and bitter cold air. With temperature this low, oxygen starts to liquify, the survivors will have to dig up a pail of liquid air, heat it up for them to breath.
The quake destroyed most underground pipelines. With no ionosphere, contacting with radio or any wireless method is impossible. The protagonist and his family lives in a obscured world far away from civilization, yet they strive to survive.
What if the Sun Disappeared - Vsauce
A pail of air - Fritz Leiber