Let's say medical science had this enormous breakthrough that allowed us to cure any illness and completely stop aging, cheaply and easily. No more death by natural causes. For the sake of this discussion, let's say overpopulation isn't an issue (we've developed interstellar travel and are filling up other planets, or we have dimensional phase shifting that allows multiple people/things to occupy the same space simultaneously and independently - or whatever).
Even though people would no longer die of natural causes, people could still be killed (accidents, homicide, war, etc.). I read a little sidebar article in a magazine a while back. It said that the chance of an adult in the U.S. getting killed in any particular year was some small fraction of 1%. Taking the inverse of that, it came out to something like 1400 and some odd years being the average lifespan if old age and disease didn't kill people.
If people only died from non-disease related extreme physical trauma, what would our perception of death be? As it is, we seem to view the tragedy of death as being somewhat proportional to the remaining years that were taken from a person. If someone dies at 60, we see it as very sad, and it doesn't seem right. But if someone dies at 30, we generally perceive that as more tragic (she was so young, I can't believe she was taken at that age, etc.). And if a child is killed, we see that as even more tragic.
If people lived indefinitely until they were killed by something external to them, how would we view death differently than we do now? How would safety concerns be different? What about war?