You can do it, but I disagree with the advice to borrow a bike to try it out. One of the biggest factors in how you will feel is the bike fit, if you were proposing walking to work, you wouldn't borrow someone else's shoes to try it out, you would look for a pair of shoes that fit you well.

I would go to a local bike shop, and tell them what you want to do. Don't let them sell you some road bike designed for racing, or a full suspension mountain bike, I would start out with a commuter or hybrid bike, but make sure that it fits you well. After riding for a while, you will have a better idea of what bike is really right for you, and if things work out, you will be going back with much more specific ideas of exactly what kind of bike you need/want.

For what it's worth, I am 53, and have an 8 mile commute each way. I ride it slowly and easily enough to not work up so much of a sweat that I have to change clothes when I get to work, and riding that way, the commute takes me 40-45 minutes. If want to push harder, I can do it in half that time.