Actually, when I was little, I had 3 goldfish in a 5 gallon tank (I know, I know, it sucks but i didn't know better at the time), who mated and had eggs. So I guess they were pretty happy/healthy.
You can keep a fancy in a 10 gallon, but obviously he won't grow, and you'll have to do 100% changes every week, or 50% 2 times a week and 100% every 2 weeks cause they poop so much it's kind of amazing. I wouldn't advise keeping a comet or long bodied type, they need much more room to swim, but a fancy just goes in circles anyway around the tank.
Water changes are key, remember! If you don't the water will be so cloudy you can't see the fish :)