In humans, supercentenarians (110+) don't tend to die of cancer - in fact your odds of dying from cancer appear to decline precipitously past the 80s and 90s. I'd wager we won't know definitively why this is the case before there are good general cancer therapies in circulation, though.
The condition that presently acts as a final full stop to catch everyone lucky enough to miss out on all of the other fun little fatal surprises at the decaying end of life is called senile cardiac TTR amyloidosis. Some description:
The short of it is "gunk clogs the tubes" but the long of it is much more involved - and like many things found in aging it has a rare youthful variant probably caused by unlucky genes. I note that the SENS Foundation puts some funding towards work on the narrow range of promising therapies for this condition:
The careful qualifications to your case above seem to catch this objection, but based on my reading that book, I would gather that supercenternarians dying of gunk not cancer aren't necessarily immune from cancer. They may just be lucky in that gunk, at this period in history, got to them first.
If it isn't luck but something else, there's also the problem of extrapolating a therapy from that. De Vere's solution to cancer was replacing every cell in the body every 7 or 8 years to circumvent the effects of telemerase ( or something like that, I'm not a gerontologist obviously.) It would be interesting if these supercenternarians's bodies were doing something like this, but it would be more interesting if it led to a more practical therapy.
First, inflammation is an important cause of cellular (and DNA) damage. The standard weapon used by your innate immune system is super-oxide, a free radical! Avoiding chronic infections is likely to be an important key to longevity, especially for those prone to inflammation.
Second, while "aging" is a natural mechanism by which your body defends against cancer, "aging" quickly is not the ideal way to prevent cancer.
Your cells will stop dividing when they have reached their Hayflick limit. This limit is not exactly caused by telomeres, but rather the lack of telomerase (a protein that lengthens telomeres). Active telomerase is necessary for a cell to become cancerous.
So the cell has its own built in mechanism to determine when it has divided too many times, as well as when it has accumulated DNA damage from external sources like UV or oxides. Your cells have "aged" when they have accumulated too much DNA damage! So aging, in this sense, is not a good way to prevent cancer. The aged body is filled with tumors that are near metastasis, and undergoes drastic changes (eg. reduction in hormones) to prevent cancer formation.
The only real way to prevent cancer is to stop cell division. This, of course, is not ideal. How would we grow to become an adult without cell division? How would we heal damaged tissue? The answer is actually in the growth pathways of mammals.
It turns out that dwarfs (little people) have very low cancer rates. See for example:
These people lack a gene (in a pathway) that is necessary for cancer to form. Knocking out genes in growth pathways in mice drastically reduces their cancer rates (the mice are also small). Want to prevent cancer? Stop growing!
This is completely unrealistic, of course. However, populations with limited activity in their growth pathways will be less prone to cancer. This extends to short people too, not just dwarfs.
Also, on a related note, there's Peto's Paradox about why cancer rates are similar across species despite vastly different sizes (mice, humans, whales).
Paper behind paywall: http://www.sciencedirect.com/science/article/pii/S0169534711...
Perhaps instead of hoping for a silver bullet to cure cancer, determine when/if your body is full of tumors, and do everything you can to prevent them from accumulating any more DNA damage.
A) Avoid situations which I know will be stressful. Not all situations that are stressful are harmful in the long run, though. For me, meeting people is stressful, but it pays itself off as if you're lucky enough to come off as likable you may have new friends, which will reduce your stress. People who socialize have been shown to live longer and be happier, according to Richard Restak in his book "Think Smart."
B) If you're troubled by something stressful (which has already occurred), I find it best to locate the exact thing that is stressing you and why it's stressing you. If it's an exam coming up, I try talking it over with people and venting. This has helped me a lot. I'll recant Think Smart again, where Dr. Restak recommends asking yourself these questions:
"Am I going to allow this experience to determine the course of the rest of my life? Or am I going to take control of my life? What can I learn from this experience?"
He goes on to say:
"Mentally reformulating things in this way will increase your feeling of control and protect your brain from stress-induced damage. This is important, since loss of a sense of control is the main contributor to the stress response. No matter how stressful the situation, if you retain control of your attitudes and responses this along will lessen your stress--even if you can't affect the situation responsible for the stress."
"Not being able to govern events, I govern myself."
-Michel de Montaigne (useful for stressful situations)
"If you are distressed by anything external, the pain is not due to the thing itself, but to your estimate of it. This you have the power to revoke."
-Marcus Aurelius (had to remind myself of this today)
Hope this helps.
The book is more about the effects of stress than how to avoid it, but the author suggests:
1 - Exercise (pick something you enjoy, preferably aerobic)
2 - Meditation (seems to help while you're doing it, less clear on persistence of effects)
3 - Control - it's nice if you have (and feel) some control in your life.
4 - Have a system of social support.
5 - Religion and spirituality. Very loose ground here.
6 - Cognitive flexibility (different coping strategies for different stressors)
Personally, I've found that regular exercise is a big stress reducer. Particularly heart rate interval training, alternating periods of over 90% max heart rate with recovery periods below 70% max heart rate.
Serious research into it is pretty young, but things like regular exercise can help control the process a bit more. We're starting to find compounds that can reduce cortisol response as well (eg. a couple of cups of black tea a day seems to noticeably reduce it.) As research progresses, we should see more advancement.
In the meantime, most of the 'good for you' stuff tends to reduce the problem. Eat well, exercise, get enough sleep, and so forth. They all work to minimize your body's response to stress. Avoiding stressful situations is ideal, but they aren't the root of the problem as much as cortisol is.
In this scheme cancer is just one example of the balance between somatic maintenance and reproduction rate.
 The disposable soma theory of aging predicts that species and cohorts in a population expecting, on average, to have high survival and low reproductive rates should invest more heavily in protecting their somas than species and populations that expect a short lifespan and to reproduce rapidly
Personally I am convinced that cancer is just one of many degenerative health issues. If you follow that line of reasoning, than the frequency of degenerative issues in skeletal remains are likely related to the frequency of cancer and any other degenerative health issue. As one example, if you simply see caries (cavities) in the remains of the teeth, that would also mean cancer was more likely to occur. (very healthy indigenous cultures have almost no cavities). Obviously this is very loose logic, and certain groups are more likely to have certain issues that are more or less likely to show up in skeletal remains, but I believe the general principle is sound.
Eat your leafy greens.
There is another simple point - who on earth will ever publish a study which proves that the whole ready food and fast food industry makes us sick, that the whole idea of office work (and driving a car!) is deeply wrong and that just vegetarian diet and regular exercise, e.g. agriculture work, and community of our equals is what we need because it defines what we are? ^_^
Of course, the study that shows that common sense has some sense will never be widely pushed through media. First, who will give you money to prove things that everyone knew but forgot? And corporations, who need to push more crap to get more profit, will stop it on very early stage, because that is the best strategy for them.
So, yes, diet, exercise and proper state of the mind (avoidance of stress, attachments, passions) together makes us healthy and younger, but people knew this for aeons without any Wikipedia articles. ^_^
1. Human stem cells will reproduce indefinitely without aging in a controlled environment.
2. Some turtles live 200 years, do they have a different DNA aging and cancer management program?
3. I thought aging was due to the polymerase having run out of buffer telomere junk data, cuts into the in-use data DNA files that are responsible for managing the refreshing of your body. src http://www.hhmi.org/biointeractive/cancer/animations.html