When is a car too old to drive safely?

An old but well conditioned car is all right to drive anytime. It's mainly the driver that makes the car safe or not.

When the frame is rusted through, I’d say it’s time to junk it. As long as you keep up with regular maintenance (tires, brakes, suspension) and all the lights and wipers work, a car should be relatively safe. But if the frame has too much rust, the car could fall apart from hitting a pot hole. It really depends where you live. In my area, rust is what kills most cars. People drive perfect running cars to the junkyard because they can’t pass inspection anymore. 

Age doesn't really matter as long as the car is mechanically sound. Naturally, you are safer in a newer car with anti-lock brakes, air bags, etc than in an older one that doesn't have those features, but any well-maintained car is relatively safe to drive.