The main conflict is that the two men can't stand the smell inside the freight car but can't stand the cold outside either. The narrator states that it is wintertime and that he left his home in Cleveland in a driving snot-storm. The train is taking them to Wisconsin, where the weather is even colder in the winters. They keep trying to do something to cope with the bad smell, but everything they try, such as smoking cigars and even building an open fire on a piece of metal, only makes the smell worse and drives them outside again. They finally resolve the conflict in a drastic way. The expressman tells the narrator:
"We got to stay out here, Cap. We got to do it. They ain't no other way. The Governor [that is, the corpse] wants to travel alone, and he's fixed so he can outvote us."
So they stay outside until they reach the next station. The narrator concludes:
We were taken from the platform an hour later, frozen and insensible, at the next station, and I went straight off into a virulent fever, and never knew anything again for three weeks.
As the title of the story suggests, the narrator has become a permanent invalid as a result of his ordeal. The moral of the story would seem to be: "No good deed goes unpunished."
the main conflict in the story is what is contained in the box, and what they are planning to do with it.