It's not the age of a car that matters, but how well she's maintained. She'll break down occasionally, any car does, but the odds of a major malfunction happening that would lead to a car crash are somewhere between 1% to 3% (I'm quoting this out of a textbook I have open right now).

As for the car running out of gas, that can't happen without you knowing about it well in advance. Every car has a display on-board that shows you how much gas you have left; even when you're "in the red" you can still drive a good few kilometers before the engine stops completely.

With driving, you have to remember that you control the car, not the other way around. Once you get used to it (and that doesn't take very long) it will give you a tremendous sense of control and empowerment. I speak from experience when I say that what I feel when driving is the polar opposite of anxiety.