How to deal with fear of failure as a PhD student?
What you experience is essentially what all grad students feel, though few of them talk about it. I would bet that 3/4 of the full professors on this list felt like that at times when they were grad students, and may still do. I can certainly say that for the first few years of being a professor, I periodically had the thought that I'm just a normal human and that the world would end if my department ever found out -- all while doing, what I believe, good work.

So you're not alone in having these feelings, though you may feel them more severely than others if you find that it occasionally paralyzes your ability to move forward. (Though, even there, I would say that all of us had weeks where we got to work, opened a browser and studied the world wide web for most of the time until it was time to go home.)

The best way to get over it is by talking to others, both your peers and your advisers (including the department's graduate adviser) about these issues. You will find that everyone, even those you think must be sooo much better than you, has these issues. I think it will help you put yourself in perspective. They may also share their own strategies to deal with things.

My strategy if I get stuck on something is to do something else -- work out the part of your thesis for which you already have results, write them up, do the copy editing -- and come back to the open questions when you're in a better place again.