Given the descriptions of the house and the area, there seems to have been a nuclear explosion, possibly arising from an accident or an atomic/nuclear war. The mechanisms of the house (and the year, 2026) suggests a society with advanced technology, thereby lending evidence that advanced weapons would exist as well. The house is deserted so the family could have simply left for other reasons. However, there are details that point to a nuclear blast. There is a big clue when the house and surrounding area are described. Note the term "radioactive":
The house stood alone in a city of rubble and ashes. This was the one house left standing. At night the ruined city gave off a radioactive glow which could be seen for miles.
The side of the house is "charred." The only spaces on that side that are not charred are the silhouettes of the five family members. They have been vaporized instantly ("one titanic instant") with the nuclear explosion.
Here the silhouette in paint of a man mowing a lawn. Here, as in a photograph, a woman bent to pick flowers. Still farther over, their images burned on wood in one titanic instant, a small boy, hands flung into the air; higher up, the image of a thrown ball, and opposite him a girl, hands raised to catch a ball which never came down.