How to deal with fear of failure as a PhD student? I am working on a mathematical topic  that is basically my own concoction. I have days where even the simplest thing takes ages to do and I realize that's because, deep down, I am terrified that things might not work out. This is in different way: I am scared my topic might not be novel enough, appreciated etc. ; I am also scared that checking parts of proofs will reveal catastrophic mistakes. 

I rationally realize that this is part of the game and I do my best to take mistakes I find as good news that my work is improving, but at times I am just paralyzed and unable to do things.

I am also bothered by the presence of colleagues in my office, afraid they might judge my work (and at the same time going to a hidden place in the library helps a little but is also very annoying and uncomfortable). 

I am sure this must be, to a different degree, experienced by many PhD students. 

What are some strategies to fight fear as you work? 

Note that I realize there really is a deeper confidence issue and I am already seeking help for that. I'd be happy to hear considerations on the possibility of therapy or counseling and how it worked for you as well as day to day tips and strategies.
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.