Thank you for the input.
Either way, I think the answer is somewhat the same, and as a technical founder, I often have to get over my own doubts as to markets, capabilities, etc. etc.
First, understand that you will be hit with stages of doubt from others and from yourself. Nothing is guaranteed, for every twitter, there are 10 blippys. You don't have a crystal ball, and neither do angels, vcs, friends or anybody else. That's the nature of the game, if you aren't comfortable with that, you can't play.
I once created an online scheduling solution for businesses. Everybody told me it was such a great idea, such a great business. It fell flat on it's face, and fast.
I also built a concert search engine, and my brother (a successful start-up founder) looked at it and thought it was useless and sucked (he told me to focus on the scheduling stuff). the concert search got quite a bit of press and was fairly popular for a while.
So that's the first part. Understand and appreciate that you don't know, and nobody else does either. Start with your eyes open.
If you are looking at market doubts, consider how many companies were first thought of as useless or flash in the pan, which became something. HP wasn't interested in Steve Woznaik's computer (which went on to become apple), why would anybody want a horseless carriage, etc. etc.
On the flip side, play devils advocate with yourself and make sure you know why you are doing what you're doing. If you only half believe in it, you aren't going to get very far. You really do need to be all in and passionate about your product, but I'd say it is more important to be passionate about the market you are targetting.
If the doubts are concerning what you are offering, you need to consider that very carefully. You didn't specify what your background is, but you can't be just an 'idea' guy. What are you bringing to the table. Often technical people (myself included) don't feel that much of the business process/sales etc. is as important as the creation of the product (development), so you may have to work hard to ensure technical people understand what you are bringing to the table, and that they don't doubt your commitment and abilities.
A big part of getting over doubts is in how you're selling. Take a good look at Simon Sinek's Ted talk to help you sell your vision. http://www.ted.com/talks/simon_sinek_how_great_leaders_inspi...
My background is in Product Management. I am good at defining product, building products, and then marketing ideas. I do however have a engineering background, MSc in EE. My idea though is about internet, social networking. I have doubts that when I first open my mouth and talk about social networking, people will just said, forget it, Facebook rules, without even listening to what I have to say. When your buyers already have a stereotype on that, it makes it very hard to pitch and now it creates doubts in me...
About your more general question "How do you deal with doubt?". The answer is that you ignore it and keep going. By definition an entrepreneur will always have doubt. He has to just keep working and getting feedback from his customers regardless of the doubt.
Accept that you dont know what your doing but devise an experiment that will make things clearer and then go execute it and learn from it. Repeat.
It's good to have doubts because they help you understand if what you want to do is really worth doing.