I would say middle school is the earliest appropriate age to start reading this story. The subject matter is very likely too disturbing for younger children to handle.

The story, for example, has constant references to the bloodthirsty ways of the lions in the veldt. The Hadley parents find Mr. Hadley's chewed wallet in the nursery. Mrs. Hadley's bloody scarf also appears on the nursery room floor. The parents fear the lions will attack them, though they tell themselves that this idea is irrational. However, the lions do attack and kill them.

A younger child reading this story could develop the fear that predators could jump from a television screen, just as the lions do in this story. Also, a younger child is more likely to experience anxiety from a story in which the parents are killed. An older child or young adolescent is more likely to have established firmer boundaries between fantasy and reality. Therefore, they would know that what happens in the story could not happen in real life.

An interesting question. The editors of the magazines Bradbury published in took care to not improperly influence the young, and their readership was quite young. Given that, and the earlier age at which children are exposed to things today, I would say early adolescence--12 to 13. That is old enough to not be frightened by what they read...but young enough to remember being like the kids in the story.