Higher organisms like we humans are made of cells, of several hundred distinct types if you exclude all of the symbiotic bacterial species that we carry along with us. The vast majority of cells have short finite life spans: they stop reproducing and self-destruct or become senescent after a number of reproductive divisions. You might be familiar with the Hayflick limit in relation to this topic: it is the number of times a cell divides before it removes itself from the cell cycle to a fate of destruction or senescence. Similarly you have probably heard of telomeres, the repeating DNA sequences at the end of our chromosomes. The length of telomeres shortens with each cell division, forming a sort of countdown clock, and too-short telomeres is one of mechanisms by which cell division is halted.
The reality on the ground is much more complex than this simple view of a cell division countdown. Some cells don't divide and last you a lifetime, such as many of those in the central nervous system. Other cells, such as stem cell populations, have their telomeres repeatedly extended by the enzyme telomerase. Different cells in different parts of the body have very different life spans, and the complex array of processes that determine those life spans is highly variable, reacting to the environment and to each other.
None of this really has much direct bearing on the life span of an organism, however. You can't just wave a wand that would extend the life of all cells, and expect to see a similar extension of life in the organism - whether that happens or not depends on the intricate details of how cells relate to organs and systems. The life span of cells is all the way down there in the depths of the machine, details internal to low-level components that are decoupled from how the machine behaves in aggregate. There is no particular reason for cell life spans to have anything to do with how long the machine as a whole can last. Some of our tissues are designed to cycle through and replace all of their cells very rapidly, in a matter of days. Other cells are never replaced and live as long as we do.
Cell behavior is subordinate to the needs of the organ or system that they are a part of. The cells of a given type evolved to have their present behavior and typical life spans because, when acting as a system in conjunction with other cell types, they produce a working organ or system that provides some evolutionary advantage. If that can be done with lots of cell turnover and short cell life spans, it will be. If it can be done with little cell turnover and long cell life spans, it will be also - but either path can produce a long-lived and reliably functional organ. This point is one that a recent article comes to eventually, after a tour of the Hayflick limit and telomere biology:
It is true that as we get older our telomeres shorten, but only for certain cells and only during certain times. Most importantly, trusty lab mice have telomeres that are five times longer than ours but their lives are 40 times shorter. That is why the relationship between telomere length and lifespan is unclear.
Apparently using the Hayflick limit and telomere length to judge maximum human lifespan is akin to understanding the demise of the Roman empire by studying the material properties of the Colosseum. Rome did not fall because the Colosseum degraded; the Colosseum degraded because the Roman Empire fell.
Within the human body, most cells do not simply senesce. They are repaired, cleaned or replaced by stem cells. Your skin degrades as you age because your body cannot carry out its normal functions of repair and regeneration.
The processes that cause degenerative aging occur at the level of cells and specific protein machinery within cells, harming their ability to perform as they should. Old, damaged cells produce more old, damaged cells when they divide. Old, damaged stem cells simply fail to keep up with their tasks of tissue maintenance. Long-lived cells become progressively more damaged and incapable, or die back, either of which causes very visible issues when it happens in the nervous system and brain.
Aging is simply a matter of damage. But how that damage translates into system failure is not a straightforward matter of cells living longer or cells dying sooner - except when it is for some long-lived cell types. Every tissue fails through the same general processes, but those processes produce a very wide range of failure modes, depending on the character of the tissue and the cells that make it up. Go beyond the comparative simplicity of the root causes of aging, and everything becomes progressively ever more complex as you move towards describing the highly varied biology of fatal age-related diseases. This is why intervening in the root causes is absolutely the best and most cost-effective strategy, the only one likely to produce meaningful progress towards human rejuvenation in our lifetimes.
As a final note, for my money, I'd wager that forms of amyloidosis are the present outermost limiting condition on human life span. The evidence suggests that this is what ultimately kills supercentenarians, the resilient individuals who have made it past the age of 110, avoiding or surviving all of the fatal age-related medical conditions that claimed their peers.