Nuclear DNA in every cell accumulates random mutational errors over time. DNA repair mechanisms are highly efficient, but nonetheless, some damage slips through. It is still an open question as to whether this damage is important over the course of the present human life span, at least beyond the matter of cancer risk, where it is well proven that more mutational damage means more cancer. Putting aside cancer for the moment, is there enough random DNA damage occurring in normal individuals to produce enough dysregulation of cellular behavior to in turn lead to meaningful levels of other forms of harm and lost function in tissues? The consensus among researchers is that this is the case, and nuclear DNA damage is listed in the noted hallmarks of aging paper, but that consensus has been challenged. This is largely a debate over theory and indirect evidence at the present time, however: is difficult to produce an animal study in which only random DNA damage is altered in degree, so as to definitively establish differences in outcome. There have been a few promising starts in that direction in the past few years, but much more work remains to be accomplished.
The cells in our bodies are constantly churning out proteins and other structures, built according to the blueprints contained in their DNA, which are crucial to supporting those cells' functions. And while in each cell most of the information contained in its DNA will be ignored, if an area of the genome important to the cell's function is damaged or develops mutations, the cell may produce misshapen proteins or simply stop functioning altogether. The effects of misshapen proteins can range from useless to actively harmful, as when neurons in a brain with Alzheimer's disease produce excessive amounts of the neurotoxic protein amyloid beta.
A few dysfunctional cells here and there don't pose much of a problem, but as more and more cells in a tissue accumulate damage over time, the health of the entire tissue or organ may be compromised. The body's normal way of dealing with such cells is to remove them through a type of programmed self-destruction called apoptosis, making way for new cells. Some cells, however, fail to die and enter a state of senescence, where they are incapable of replicating but are still left to take up space in the tissue. But especially dangerous is the case of a cell with damaged DNA that doesn't enter either apoptosis or senescence: damage drives mutation rate up each time the cell replicates, and if a mutation provides a survival advantage or switches off the cell's protective mechanisms against tumor formation, this can eventually lead to cancer. Cells that divide frequently, such as skin or lung cells, are most susceptible to this danger.
Our bodies may have evolved an impressive variety of damage repair mechanisms, but with increasing exposure to damage-causing agents in the environment and the damages caused by our own internal processes, compounded with the declining effectiveness of our protective mechanisms over time, DNA damage and mutations are bound to accumulate as we age. Some evidence suggests that caloric restriction may mitigate these effects. However, since no drugs yet exist that will prevent or repair DNA damage, all we can do at the moment is try our best to avoid harmful agents like excessive sun exposure and smoking.