Those people are called Laron dwarves. They have a short height because of a mutation in a growth hormone receptor protein. This mutation prevents the liver making IGF-1 (insulin-like growth factor-1).
IGF-1 is some sort of growth hormone that makes the body grow. If the body has a deficiency of it growth is stunted, like in people with Laron syndrome.
However, Laron dwarves hardly contract cancer or type 2 diabetes. They seem to be almost completely immune to these diseases, which points to a role of IGF-1 in cancer and diabetes.
Other research has indeed shown involvement of IGF-1 in cancer and diabetes. The more IGF-1 circulating in the body the higher the risk of cancer or diabetes. Other studies show a link between IGF-1 and longevity. Lab animals like worms without IGF-1-receptors even have a double life span.
Conversely, the more growth hormone, the more IGF-1 you make and the faster you seem to age.
Ironically, growth hormone is touted as a powerful remedy against aging on the internet and in many health books, despite studies showing a clear link between growth hormone, IGF-1, cancer, diabetes and accelerated aging.
Nonetheless, Laron dwarves provide scientists with yet some more tantalizing insights into the why of the aging process.
This is a movie about Laron dwarves and aging: