Here are two quotes from writers of antiquity, letters in a bottle to demonstrate that for all the great differences in time and place, the core of human nature in complex civilizations remains much the same. So far, at least:
Death: There's nothing bad about it at all except the thing that comes before it - the fear of it.
I think the slain care little if they sleep or rise again.
Insofar as death is oblivion, the destruction of the self, it is rational to be unconcerned about being dead. You won't exist to have feelings on the matter - which is exactly the same situation as for all time prior to the point in your development at which you like to think that you became yourself.
Equally, it is rational to be very concerned about being dead at some point in the future. Humans are creatures of action. We like to achieve, observe, and experience. We place value upon these things, and death will stop us from gathering that value.
So the world of people might be divided at any point in time into (a) the group that is horrified by the prospect of oblivion, and (b) the group that is unbothered by personal extinction. It's a very sharp dividing line, not often explored in casual conversation, for all that one person's views might cross back and forth between camps over the years. Try a poll of the folk you know at some point in time: I think you'll find the results interesting.
Whatever your opinions on oblivion, however, it seems near universally agreed that the process of becoming dead is something to be feared - so much so that we work to hide the ugly reality from daily life:
Are you afraid of getting older? You should be. Degenerative aging isn't pretty. That said, we live in an era characterized by a fascination with youth; aging and the old are put to one side, and the ugly details of the way in which the body and mind break down are glossed over or shoved under the carpet. Move on a step from that and you'll see the bevy of folk trying to sell you the message that aging into frailty and death before you're ready is just fine - that you shouldn't worry about it, that you should just relax into your life being taken from you, one piece at a time. But those talking heads are spouting nonsense. You should absolutely be afraid of aging.
Terrible pain, injury, and degeneration are not on anyone's wish list. But we'll all be receiving these dubious gifts anyway - unless something is done about it. The difference between our age and the age of the Greek stoics is that we have the chance to do something about it: defeat aging by repairing its damage, improve our biochemistry, and ultimately replace our bodies with superior technology that is immune to all that plagues us now.
There is a school of thought that suggests selling the concepts of radical life extension as indefinite postponement of death is inefficient. In this view, half the people the world don't really care about the future state of being dead; what they care about is the prospect of pain, injury, and degeneration. Therefore the better pitch for engineered longevity focuses on elimination of the pain and degeneration of aging, and the continuation of youthful health for as long as a person desires.