A commenter wrote:

If you only had 20 minutes, what would you do to convince an intelligent (college educated or professional) audience of the significance of life extension beyond 120 years? Assume they do not commit the Tithonus error and are rational enough to understand that probably do want to live indefinitely so long as the quality of life meets their own standard. The problem then becomes convincing others that it is indeed possible to live very very long lives. However, how can you convince someone that extreme life extension is scientificially valid research? It seems like some persons will dismiss the possibility because there is no such empirical example, just as planets outside our solar system were not outright denied, but ignored.

Given twenty minutes, it is really only possible to set forth an outline of a position, and show people where to look for the supporting evidence. But an outline might look something like this:

• Consider that there is a very wide range of life spans between even very similar species. Take the naked mole rat, a mammal that lives for nine times as long as similarly sized rat species. Or the larger whales, mammals that can live for nearly two centuries. Very long-lived mammals are possible at any body size, and their longevity is a matter of their particular evolved genome and metabolism. More esoteric species are even more long-lived: four centuries for some clams, for example, and no one knows how long lobsters and some urchins can live because they appear to be essentially ageless.
• But evolution has clearly not selected for extended healthy life in many species, a broad range of mammals included. This is demonstrated by the fact that there are now something like twenty different ways of extending mouse life span by 10% to 60%. Many of these are modest genetic engineering projects: a single gene, or few genes altered, well within what we'd expect evolution to accomplish on its own.
• We should not expect primates, humans included, to be exceptional in this regard. While we are long-lived in comparison to many mammals, we are far outclassed by a good many other species.
• An animal such as a mammal is a complex system built of many interacting subsystems, which are in turn built out of many redundant parts. The general behavior of systems of this nature is well described by reliability theory: mean time to failure grows small as a system ages because it accumulates damage that knocks out its redundant components. Aging and all that comes with it - frailty, disease, organ failure, and ultimately death - is the result of accumulated damage and the flailing of damaged systems.
• At the lowest level, we are machines: our cells are finely turned, reactive, programmable, self-repairing machinery. The nuts, bolts, cogs, and pistons are proteins built according to the patterns encoded in our genes. Over time cells build up damage in the form of broken, malformed, or unwanted proteins, some of which can be repaired or removed, and some of which cannot. This damage is a natural consequence of the operation of our metabolism: it is accumulating slowly within everyone's body right now. Everything that happens to us in aging ultimately stems from a build-up of broken, worn, misplaced, and errant parts of our biological machinery.
• All means of extending longevity demonstrated to date in mammals in the laboratory are essentially forms of damage reduction. They reduce the rate at which damage accumulates over time: by creating greater damage resistance, attenuating the effects of damaging processes in our metabolism, or spurring cells and the immune system to greater natural vigilance, repair, and recycling efforts. Less damage per unit time is exactly a slowing of aging.
• But if slowing down the build-up of damage is good, how much better will it be to repair and remove that damage completely? Consider that the only differences between a young body and an old body are (a) levels of damage and (b) the disarray of biological systems and organs reacting to that damage. Complete repair of damage is one and the same with rejuvenation.
• There is a fortunate confluence of purpose between research to repair the biochemical damage of aging and research into therapies for age-related diseases. These diseases are caused by the varying forms of biochemical damage that create the condition that we see as aging; a therapy that repaired a specific form of damage would be beneficial for everyone who suffered from the diseases it causes. It would lift some of the burden of aging, restoring a body some way towards its youthful state, and this would be beneficial for everyone who is old.
• But we do not have to be hypothetical when talking about the forms of biological damage that cause aging, or ways to approach the development of therapies that can remove that damage. The damages of aging are in fact well documented in many of the varied life science and medical fields, and are summarized in the Strategies for Engineered Negligible Senescence:

Some tissues lose cells with advancing age, like the heart and areas of the brain. Stem cell research and regenerative medicine are already providing very promising answers to degeneration through cell loss.

We must eliminate the telomere-related mechanisms that lead to cancer. de Grey suggests selectively modifying our telomere elongation genes by tissue type using targeted gene therapies.

Mitochondrial DNA is outside the cellular nucleus and accumulates damage with age that impairs its critical functions. de Grey suggests using gene therapy to copy mitochondrial DNA into the cellular nucleus. Other strategies for manipulating and repairing damaged mitochondrial DNA in situ were demonstrated for the first time in 2005.

Some of the proteins outside our cells, such as those vital to artery walls and skin elasticity, are created early in our life and never recycled or recycled very slowly. These long-lived proteins are susceptible to chemical reactions that degrade their effectiveness. Scientists can search for suitable enzymes or compounds to break down problem proteins that the body cannot handle.

Certain classes of senescent cell accumulate where they are not wanted, such as in the joints. We could in principle use immune therapies to tailor our immune systems to destroy cells as they become senescent and thus prevent any related problems.

As we age, junk material known as amyloid accumulates outside cells. Immune therapies (vaccines) are currently under development for Alzheimer's, a condition featuring prominent amyloid plaques, and similar efforts could be applied to other classes of extracellular junk material.

Junk material builds up within non-dividing, long-life span cells, impairing functions and causing damage. The biochemistry of this junk is fairly well understood; the problem lies in developing a therapy to break down the unwanted material. de Grey suggests searching for suitable non-toxic microbial enzymes in soil bacteria that could be safely introduced into human cells.

• It is not so hard for people nowadays to visualize the transformative power of regenerative medicine. This is a field in full swing, to the accompaniment of large-scale publicity and public support. There is the ability to renew aging cells, such as nerve cells in the retina that are never naturally replaced and slowly give way to cause age-related blindness - a way to give a renewed life span to small but vital bodily systems. Or the ability to grow and transplant completely new organs to replace age-worn hearts, kidneys, livers, or lungs - but this just scratches the surface of what will soon be possible.
• Yet regenerative medicine is one field of seven in the list above. Each of the others has the potential to be just as transformative to health and longevity. By removing the damage that degrades a range of bodily systems, a body can be moved closer to the state it was in when young. Systems that were flailing and causing problems of their own can be restored to a healthy state, and return to contributing to health rather than damaging it.
• You don't have to take my word for it, of course. Regenerative medicine and organ building are thriving growth fields, populated by thousands of highly skilled researchers who are well aware of the benefits their work will bring to the elderly and the damaged. In labs around the world, and to a lesser degree, work is slowly proceeding on each of the other other six fields above.
• Within the broad life science research community it is now taken as a given that extending healthy longevity is serious, meaningful science. Billions of dollars have already been invested into research and development for early-stage longevity science. The arguments are over how it will be accomplished, and what is possible to achieve within our lifetimes.

That seems to be a decent first pass at the topic. It needs refinement, but you're all welcome to have a go at it.