The possibility of whether eternal life would work on earth, is an idea which has been pondered, probably for centuries. It certainly has been tossed back and forth during my lifetime as can be observed in some of the movies, discussing the possibility.
Sickness and pain would have to be somehow deleted from life - an idea extremely attractive to a society which sees no benefit in suffering.
Assuming that, there is the movie, "Tuck Everlasting". Tuck drinks from a spring in the forest and finds he does not age at all (aging also has to be removed from the equation). He does retain his youth and strength, but watches all his loved ones live, get old and die. The movie ends with him standing all alone.
In fact, loneliness is something humans feel would be a given, if we outlive those we love. That even can be seen in the elderly who have outlived parents and kids - they are extremely lonely and loneliness can be more painful than physical pain.
Which brings in the question, is physical pain the worst thing we can suffer? According to a Dr Oz show, the consensus of the audience (and it seems, Dr Oz as well) was that if the physical pain gets pronounced enough, it would be a logical solution to seek someone to help us end our lives. A woman with a type of muscular dystrophy which has taken away her ability to walk and use her arms, was prominent in the show - she did not use a respirator so apparently could eat and breath well and also could move her head fairly well. She was depressed and wanted the right to have a physician help her end her life. Interestingly enough however, when Dr Oz asked her if she would favor not being treated if she got pneumonia again allowing her to die, she evaded the question. (She has had pneumonia twice and has been treated).
Another man, who looked elderly and was much more disabled than the lady - he required a respirator - told the audience that he appreciates every day of life he has been given.
Interesting, are the movies, "Invasion of the Body Snatchers" (one made in the 1950's and one made in the 1970's). Tonight I saw yet, another remake entitled "Invasion" released in 2007. The story is that some type of invasion from Outer Space takes over humans and recreates them as devoid of emotions. "There will be no more war or suffering," the protagonists are told. In all of these movies, there are a couple of humans who fight against the takeover, preferring their flawed emotions to becoming a being who is incapable of human love. The latest version has a happy ending. A vaccine is found and given as a sort of antidote to the cells taking over humanity and everyone is cured and life goes back to normal. But the leading lady is shown in the last scene as wondering whether the world with war and sadness is really preferable.
In the second and third movie of this group ("Invasion of the Body Snatchers"), each human who is taken over, is shown going through a type of aging after which s/he wakes up with no emotions, no ability to love. This seems to be asking the question - if eternal live exists, wouldn't we have to be changed? Our human concept of the change necessary, is being given - what the psychiatrists calls a blunted affect, i.e. a deadening of the emotions and all we care about.
Utimately, there seems a consensus that an eternal life here on earth would not be nice. The movie, "Cocoon" depicts elderly folks stumbling across aliens who have somehow achieved eternal life but in order to obtain this, these elderly must go with the aliens to their planet, far away from the earth.
The Christian concept of eternal life is not a lessened awareness but rather a heightened awareness. Something we humans cannot picture. And even if we do not understand how this will be achieved, we have the testimony of Jesus, still the only man who ever made such an impression on the earth that He literally stopped time (which we then, divided in "AD or "anno Domini" i.e. "after God" and BC i.e. "before Christ"). If we still have questions, the Bible tells us "eye has not seen, ear has not heard, of the glories which God has planned for those who love Him."
There is a lot of evidence, observational and logical that we should not try to force our premature death but instead should patiently wait for God to call us. There is also a rather impressive body of evidence that when God calls He will "make all things well" , not by us ceasing to exist but rather by our living in a full beautiful manner, the likes of which we cannot concept here on earth.
And of course, Cardinal Dolan said it much better than I did! :)