What is killing people in this movie?

The radiation sickness was symbolised by shadows around the eyes. The film wanted to keep things low key, without being overly graphic about the nature of radiation sickness. The audience did get to see some diarrhoea and vomiting. That was sufficient enough.

As the late Roger Ebert noted, this film doesn't even show us a mushroom cloud. He still enjoyed the story.

They start getting sick by looking bad - and the mom barfs in one scene -- in another scene she runs her fingers through her hair and a few tufts come out. The little boy was bleeding from his butt.

When the boy said there is sand on my plate, that is an indication of fine particulate fallout. No one in this film took any countermeasures. Unless it was the gamma rays, but there was no blast or overpressure damage and SF is hilly so I'm thinking it was fallout from SF......in a way their father was with them

radiation sickness and people getting sick for other reasons but now (after the war) there is no way to take care of them. Anyone on medication would be in big trouble once their pills run out. Pneumonia or the flu would be big killers too.