Best Answer - Chosen By Voters: Yes, it probably is "Flat Earth" thinking. Your question may very well be beyond the average human range of comprehension at our current level of being (i.e. -- we can't even figure out how to stop killing each other!).   Should we ever be able to raise this level, life as we know it would be quite different and THEN, perhaps, we could entertain the possibility of physical immortality . . .  
My Response:  Good Point.  And the recognition that our real self-interest and the true interest of the other are identical will take us to that higher being.  The universe is simply waiting to reward us for manifesting this recognition in OUR DEEDS.. 
Stupidest Answer:  Immortality is fundamentally impossible!  Live with it!   
My Response:  Again, if you can't be sure, why are you expressing such certainty?  It is really dishonest to do so.  As I suggested, you are most likely defeating yourself with a delusion of certainty.  Sadly typical.
Socially Responsible Answer:  It would play havoc with an already over- burdened Social Security system! 
My response:  Not if we raise the retirement age in accordance with the average life-expectancy.  And besides, that is a ridiculous reason to keep from finding out how immortality might be achieved.  What's more, physical  immortality will provide the ability to endure long space voyages to other inhabitable planets.
Fake-Wise Answer:  I once had a friend who said that "The only reality is what we perceive as reality, there is no ultimate reality, etc..."  I then pushed him into a mud puddle. I asked, "Are you wet? Does it depend on how you feel about it, or how you percieve it?"  He said, "No".  I said, "That is reality."
My Response:  Give me a break, socrates.  Im not your stoner friend.  I'm not suggesting that there is no ultimate reality, but rather that reality may not be what you believe it is today.  Is that so far-fetched?  We are in the Dark Ages if you compare where we will be in 200 years.  History has shown us that.  200 years ago, we thought human fleas were a fact of life that we had to 'live with.'  Open your EYES, your arrogance is lethal!!!
The Defeated Answer:  All I can say is that, in all of the years of human history, no one has figured out a way to "out-think" death. That, in my opinion, means that death is reality, probably the ultimate reality, the only reality that we can really depend on. 
My Response:  In all of human history up to very recently, nobody was able to "out think" gravity.  Yet the human race eventually found a way to build a flying machine.  Sorry, your reasoning is flawed.  Let's not assume the past dictates our future.  Let's not close our minds to something great because the past never knew it.
The Hmm answer:  Acceptance of reality is not self-defeatism. While it's true we do not know what the future holds, it is highly unlikely science will ever find a cure for death. From a religious perspective, death is a "wage" of sin and is, in fact, by design. Only The Designer can change that. Will He? Nobody can answer that question. 
My reponse:  I am asking if your belief really reflects reality.  You don't know that the designer has not built the possiblity of human physical immortality into the design.  So why, having no knowledge, assume that the designer has not done so?  Get off of the flat earth, my friend.  Humility to seek for answers is key here.
The Native American Answer:  Immortality IS the ultimate reality, but not in the way most people think of it. Energy does not die, it simply changes form.  As we now know, we are all of us entirely made of energy.  Our fundamental elements are ageless.  Atomically we have been parts of clouds and plants, dinousaurs -- all manner of life and universal matter.  Everything is in a constant state of change.  To think that one person can live forever in that one particular blueprint of life, is to deny the ultimate reality- the one and ONLY constant...CHANGE!  Identify with the fact that you are part of the eternal ocean of energy and it will bring you both enlightenment and immortality.
My response:  Good point, but not good enough.  While what you say is true, it avoids the question I'm asking.  I am asking whether or not it is possible to keep our fundamental elements intact for as long as we want them to be.   Do we not have a duty to stay intact for as long as we possibly can, regardless of what happens if our ageless molecules are somehow dispersed by accident?  To understand the inadequacy of your response consider how you would feel if I blew up your loved one, and then fed you back your answer.  My hunch is that you will be far from consolled.  Your assessment of reality does not free us from the obligation to try what I am suggesting here.    CYCLE OF LIFE CYCLE OF DEATH