$60 is a wonderful start! When I was I starting out, I only put away $20. It was all I could afford at the time.
Yeah... money was that tight back then. But after that, I kept adding $20 to it with each paycheck. Oh yes, I remember screaming wildly, celebrating as a hundredaire.
Anything you can put away will give you that much more of a piece of mind, and as it slowly grows, the larger it is, the more it can protect you from financial setbacks. But I think the most important thing is to get into the habit of doing it, and keeping at it.