However, I did eventually get (mostly) over it, and the prospect of death doesn't really frighten me much any more except in the way Dr. Peikoff discusses, which I take to be a reasonably rational approach to the whole business. So how does one go about getting over such a potentially paralyzing dread?
I think that death seems particularly horrible to young people who are living their lives primarily in the future--they may have many things they want to accomplish or big, nebulous, long-term goals, and there just doesn't seem to be enough time to get there no matter how long they live. In addition, there may be an aspect of finding one's current circumstances nearly unbearable, so that one feels that any goal is, at best, a long-term, far-away thing.
Well, the "cure" is to stop thinking like that and go get involved in something productive NOW. I'm not saying you shouldn't think about the future, but instead of thinking of it in terms of "well, thirty years from now I hope to have accomplished X, Y, and Z", think about it more in terms of "today I'm going to do this one thing that is moving in the right direction". Or several things, so that you can look back at your day and see something that you accomplished rather than dwelling on some impossible future to come. You have to get involved in and enjoy the things you are doing now rather than only dreaming about the results that you may get many years from now. It can be hard when you're young and your life is a big complicated stressful mess that you're struggling to get on top of. But you can do it, and it at least eliminates one source of irrational self-induced stress in the process.
Also, if you're experiencing this kind of fear, it might be a good time to take a look at what you are doing and figure out whether it's actually leading you where you want to go. There are a lot of things that people will shove on you as some kind of self-evident necessity (college, grad school, getting married, having kids, whatever) that may not suit you in any way whatsoever. So don't concentrate on the prospect of death as a way of ignoring what's really bothering you, currently.