Well, here's good news! It turns out that, yes, indeed, if we eat a high-fat, moderate-protein, low-carbohydrate diet--the kind of paleo diet that allows us to burn fat rather than sugar as our primary metabolic fuel — we can live to be as old as a Struldbrug! In other words (if you haven't recently read Gulliver's Travels — lol), a very long time!
In a January 2014 paper titled Ketone Bodies as Signaling Metabolites, the authors describe how, when we're in a state of nutritional ketosis, we produce β-hydroxybutyrate (BOHB) as our major ketone. BOHB then turns around and via some cell-signaling pathways is thought to "culminate in enhancement of longevity in mammals." In other words, we'll live to be as old as the hills if we we eat a bunch of fat, keep our carbs low, and don't eat excess protein (which our body can turn into sugar). Moreover, this mechanism is neuroprotective, sparing of the protein-responsive mTOR pathway (hence the longevity), and has various other benefits, which you can read about. Here's the link to the paper, with many thanks to Craig Bates for posting it.
If you want to know more about nutritional ketosis (and how it's different from ketoacidosis), you can read my post about it here.
For a wonderful lecture on the subject, check out Carbivore to Carnivore: Taming the Carb-Craving Monster
And, if you really want to dig in, you can check out the book The Art and Science of Low Carbohydrate Living, either at the website of the authors, Stephen D. Phinney and Jeff S. Volek.
(Photo credit: Here we observe cell signaling at the microscopic level/Artist Ben Arthur, video still from NPR's The Invisible Human Microbiome)