I couldn't sing or play when I knew anyone was listening until three years ago.
Pick a VERY supportive friend whose judgment you value and perform over the telephone a few times. It gets pretty silly pretty fast, and you end up inviting him/her to come listen in person. Then, when you're comfortable with that, add a few more friends to the gathering (like 2!) and play just one or two songs for them, and then hang out and do something you all enjoy. Finally, when you're comfortable, tell them you want to do an open mic, and get them to come sit in the front row. Focus on them, and break a leg!
My friend Lisa badgered me into playing for her on the phone when she was sick, then had to hear me in person, and finally, she, and my brother, and another friend convinced me to do an open mic.
Now, I love performing, and actually play for money!