No, it's not plausible. There are four main issues: Heat retention, thermodynamics, too much oxygen, too little water, air retention.
Heat retention is an issue. The basic problem is whether any plausible amount of coal burning will produce enough heat to replace the heat lost through leakage. I've worked with liquid nitrogen and liquid helium and the 200 degree centrigrade temperature difference between room temperature of LN2/solid N2 outside. I'm pretty sure that no plausible amount of casual insulation would retain enough heat for the kind of fire described in the story to keep the place warm. (Remember, it only takes a pail of air now and then to provide oxidizer.)
(If someone were so moved, they could easily calculate the heat produced by burning a pail of solid oxygen/hour and make an estimate of the size of the room and then compute the R-value of the walls needed to retain a livable temperature. I'm too lazy...)
I have further doubts about the thermodynamics of the burning. Assume that the insulation is perfect. There is still a lot of air lost through the flue, which carries away the combustion gasses but also carries away air. Can a bucket of air every few hours provide the oxidizer to burn coal and to replace the air that leaks out through the flue and the door? I strongly doubt it.
Speaking of air, as I recall they "mine" pure oxygen. A pure oxygen atmosphere will kill you if you live long enough after the coal and everything else flammable in the shelter catches fire. A pure oxygen atmosphere creates a deadly fire hazard -- as the Apollo 1 astronauts found out. And if any LN2 is formed, it can cause porous, flammable substances to detonate.
Water's another problem. They have a good source of oxygen and of carbon, but no source of hydrogen. If the shelter air contains any water vapor -- and it must for health -- that water vapor will escape with the flue gasses carrying away precious hydrogen. The shelter will pretty quickly desiccate.
Finally, the elephant in the room (in the shelter?) is air retention. Maintaining a breathable atmosphere against a vacuum requires good seals. The description of the seals they have is a guaranteed, continuous, substantial leak. (Which would substantially exacerbate the other problems!)
Bottom line: Not a chance. (But it's still a great story!)