It's easy to show that the Luthers are poor, especially after the father dies and the children, who have promised to do so, have to survive on their own in the absence of any social welfare aid. Their home is a shack, described as "woeful" and "seedy." It is also "downright disgraceful." The children wear ragged clothes. Things get worse in winter, when the roof of their shack collapses and snow and animals can get in—and when they have to bring farm animals (and their odors) inside to live with them. As if that is not bad enough, the children end up having to move to a cave.
Riches are harder to find in this story of poverty, but one example is in the North Carolina mountain nature all around them: the area is described as "the fairest land of them all." As long as the weather remains good, the children can live off the land through the wild plants they gather. They can also feel rich in the beauty that surrounds them: they might live in a shack, but they can feast their eyes on such loveliness as mayapple, ginseng, goldenseal, and queen's delight.
Further, they are rich in grit, resourcefulness, and a strong sense of humor which helps them survive all that plagues them as well as a strong connection, even as they quarrel, to each other.