Something that was mentioned, but hasn't been discussed much here: Death (or, rather, the awareness of it on an unconscious level) is what drives human beings in their ambitions, to strive, and overcome. Without that fear, the danger of stagnation for the majority would greatly increase, and for those who are highly sensitive, it would tend to become completely unbearable. And Pointfinder, I think you haven't really thought this out. Immortality? Yes, you could have those experiences and enjoy them for a long time, perhaps even centuries. But true immortality would soon leech things of their savor. The very fact that things are ephemeral is what makes them so precious. If they are endlessly repeatable, they lose that quality -- just look at any behavior that is that way, and how human beings tend to react over a long course of time with that behavior: They keep trying to make it more exciting.
And, if only some people were taking the pill, then they would watch those they are around wither and die. Now, someone who is totally self-absorbed, and lacks empathy (we're talking near sociopathic level, here) might not be affected by this. But most would either become calloused, or those losses would begin to hurt far too much for you to ever care for someone again for fear of going through that inevitable loss again, and again, and again.... Either way, the emotional connection to other people would almost unavoidably be damaged, permanently.
And, for those who love live, it would be even worse, as most would begin to find themselves more and more afraid of taking chances, and perhaps encountering fatal accidents or the malice of others, etc., the longer they live the more they'd be likely to become encysted as that life became more precious than the adventures they would experience by becoming involved, at the risk of losing that life. (Or having it damaged -- immortality as a paraplegic? Perhaps irreparably, no matter what medicine achieves? There are likely to be such, at very least for a long time to come. How long would you be willing to be in that position? Or totally paralyzed, not even able to speak? Would you be willing to risk this by going out on these hikes, or rock climbing, or even driving in the increasing rush-hour-traffic?)
And the population problem: As it increases, so will crime due to displacement -- heck, that's going to become worse and worse anyway, as automation makes more and more jobs obsolete without us getting our heads out of our butts to make use of the human potential of human beings! Imagine what it will be like as the population becomes so massive that our resources are depleted within a quarter century! (As I've mentioned elsewhere, just in the 30 years since I've graduated, the world's population has close to doubled. In the last century, it's close to tripled, if not more than, and this in a century where we've managed to kill ourselves off in greater numbers than perhaps any time in history. Mass enforced sterilization, anyone?)
Even longevity presents us with these problems: As our lifespan has increased, so has our unwillingness to take risks, overall. People want things safer. (Yes, there are the danger sports and such, but the number who engage in those is incredibly minute, compared to the everyday risks that we used to face without giving them a second thought.) Heck, they won't even let their kids play on playgrounds without special padding or turfing anymore in many places! And fewer and fewer places will you see people letting their kids get into the (quite normal) scuffles that children have always engaged in as part of their socializing process. Not fun, but one does learn valuable lessons from it for dealing with other people throughout life. Without those experiences, people are less and less able to cope with others. Can you imagine how less likely people will be to risk their children having any injuries in such a scenario as immortality, especially if only a very few are allowed to have children -- say, to match the number of people who die through serious disease or accident within each year?
And as for getting us out into space -- if we're less willing to take risks here, what on earth is going to goad people out into the most dangerous environment of all, where there is no room for error? Population pressure? More likely we'd start enforced euthenasia before we'd risk that. Fear is an incredibly powerful force, especially when people don't acknowledge what it is they're afraid of. The monster is always more frightening when you don't allow yourself to see it, you know. Eventually, we might start such a drive ... but once it gets going, you'd have different camps: those who feel that this is unnatural and against God's plan and who resist it; those who want to go out there and colonize and are actually willing to take the risk, but aren't fit (for whatever physical or mental reason) to go, those who are but are afraid to go, those who can go and are willing, but will be severing ties with all they know and so pull back for that reason... and the few remaining who are both fit and willing and able to make that final emotional adjustment that they will be leaving Earth forever, along with everyone and everything they've known and loved, to never encounter either again (one hell of an emotional readjustment to make!). And those who are found wanting are quite likely to fight the whole thing viciously, including with riots and bloodshed.
No, thank you. Death is our goad to actually grow as a species, as well as individuals. There may indeed be the extremely rare individual for whom immortality would be a blessing ... but I'd venture to say that they are considerably less than one in a hundred million. For the rest, it would rapidly become a living hell.
Personally, I would like a bit more longevity (I think), in order to see a few things come to fruition, to learn more, and read more, to see places I've always wanted to see but not been able to. But immortality, no. That's a fate that makes the Mariner's albatross seem a puny punishment indeed.