How Do You Say No To Someone When They Ask You Out Without Hurting Their Feelings?

Very very tactfully. Especially if it is someone you associate with on a regular basis. You can try telling them that you aren't interested in going out with anyone at all right now. But then you have to be careful about then going out with someone anytime soon.  The best approach is honesty. Tell them that you think they are a great friend (if that is the case) and that you don't want to do anything to harm the friendship. And then make a really concerted effort to maintain the friendship.
This is one of my least favorite situations. Especially when it's someone you really like--but just not in "that way". Good luck, I hope it goes well.

It is my personal opinion that there is no way to tell someone know without hurting their feelings.  Although the words have left his mouth that he's glad you're "just friends" there may always be that part of him that wonders what could have happened.

Not replying to guys texts and calls