There are various ways of shuffling off this mortal coil, but people actually die from injury (such as a fall or car accident) or disease (such as cancer). No one dies of old age. Usually when a person is said to have died "of old age," it means that he or she succumbed to one of the diseases common in our later years.
While many older people do suffer from health problems, disease does not automatically accompany aging, and seniors are living longer and healthier than ever. Just as youth does not guarantee good health, old age does not guarantee poor health.
It is true that living cells have a finite life span, but that doesn't mean that the organism simply dies because the cells are old. Instead, genetic mutations, diseases, and damaging effects of the environment can foster a specific disorder or disease. As people get older, their cells simply don't work as well, and can't stave off disease as easily or heal as well as they once could. As a result, older people may die from injuries or diseases that a younger person would easily survive. But nothing dies from simply being old.