There's not just one race between lifespan and technology, but several, depending on how much money you have. The 20th century witnessed the development of major life-extension technologies--primarily antibiotics and sanitation--but those technologies are still not accessible to the majority of people living today. The next generation of life extension technology is also going to be accessible only to a limited portion of the population--in all likelihood, a much more limited portion than the first. It's entirely likely that some humans living today will have access to technologies that will extend their lifespans for centuries or more, but most of us probably won't be able to afford that kind of medical treatment.

See Green Mars by Kim Stanley Robinson for more on this issue.