I’d like a relationship but I don’t like sex

I’m a 24-year-old girl and have never been sexually attracted to anyone. However, I would like to experience a romance. 

I’ve had a couple of relationships that involved sex and they didn’t work because being sexually intimate turned me off.

Friends have suggested it could be because I’m gay but I’ve never been attracted to a woman either. How can I find a romantic partner who will accept me for who I am?