The only place I've ever felt at risk was walking down a really nasty street in Amsterdam - ironically, on a Sunday morning, and I was trying to get to church!
Obviously stuff sometimes happens (I was being followed/hassled by a slimy guy), but that could just as well happen at home, and the way you handle it is more or less the same. Be smart, be aware, look confident, and you should be just fine. Think about all the millions of women who travel all over the world every year, and compare that to the number of horror stories you hear. The odds are in your favour anyway, and you can up them with just a sprinkling of common sense.
When I travelled on my own, I found that I met tons of people, and had people to hang out with whenever I wanted.
As a personal policy, I do avoid war zones, though...