To say that death isn't inevitable isn't the same as saying it's particularly likely that any of us will evade it, but the chance is not literally zero, and by my own estimates, I say it's non-negligible.
It's pretty uncontroversial to say that life extension technology is possible. There are times in the past when people didn't live as long as they now do. Finding more effective cancer/Alzheimer's/heart disease treatments is life extension. It's within the realm of possibilities that life extension tech will advance enough to let us live for some centuries. Cryonics (freezing yourself after death) is also something that could possibly work. It's possible that in the future, we'll figure out ways to successfully thaw and revive people who undergo cryonics.
Neither of these are immortality, exactly. For that you would need some sort of mind-state backup system or "consciousness uploading." That tech might not be available for a long time, but life extension/cryonics could hold many people over until it's available. Many people believe that it's either impossible, or that it wouldn't "be you" when you were restored from a backup. I think that these positions are wrong, but regardless, anyone who takes such a position has to allow that there's some possibility that they're wrong, and in fact such tech is possible and would work as intended. The theories of personal identity argued in Reasons and Persons by Derek Parfit are relevant here.
I think it's important that we don't accept death as inevitable, because if people don't hold out hope of conquering death, then it's less likely that we'll conquer it in time.
I never struggled with an intrusive fear of death, so I don't have detailed suggestions for how to overcome it. But facing up to the fact that you are probably going to die, and introspecting into what specifically makes you so afraid, then taking measures to address it — that sort of thing — will probably help.
Measures that might help, depending on the nature of one's fear:
- Starting a mindfulness meditation practice to learn to notice when you fear death, and learn to dismiss the fear
- Seeking psychotherapy with a similar goal
- Study philosophy. It's possible that your intrusive fear of death stems from contradictory beliefs you hold, and you just haven't noticed the contradiction.
- Registering for cryonics, so that you know that you have done something to fight back against death (this might well increase your fear of death, beware)
- Switch careers to work on life extension technology. When vikings raid your village, you will be afraid. But the nature of your fear will depend on whether you decide to cower in fear or take up arms and fight back. Maybe you prefer the fear felt by someone fighting back.
But remember that death is horrible, and it's right and proper to fear it, as long as the fear isn't too intrusive.