i doubt we can achieve eternity, we've been able to extend the life of most people, even in the 3rd world people generally have the ability to live longer than the people of 1000, even 500 years ago but we haven't been able to halt bodily degeneration, we get to around the 60 mark and things just begin to shut down, liver, kidneys, lungs, heart, bowels, bones marrow, it all reaches a point where the rigours of an extended life and prolonged damage cause them to cease functioning at a great enough capacity to sustain our lives. not to mention mental function as Stretch mentions, iirc scientists suggest the longer you live the higher the likelihood of developing alzheimer's and there a a host of other mental conditions.
it wouldn't benefit us either, look at the problems we already have with unemployment, over crowding, over populated cities and people literally living on top of each other, we'd need to limit growth to near non-existent (people are still going to get hit by a bus) and then we'd need to control who could give birth. perhaps on the people who can afford to buy the pre-requisite license. not withstanding what you'd do with all those extra (infinite?) years, if you spend your life slobbing you'll continue spending your life slobbing, it's hardly going to be an enriching experience for many people.
i'm happy with my 50-90 year lifespan (even if i am potentially nearly half way through it), i've still got plenty of time ahead of me to enjoy my future, and i've had many good, and bad experiences in my past, a vastly extended lifespan would mean things would have less meaning to me at least.