As other people have already mentioned, you can keep one fancy (i.e.: double-tailed) goldfish in a 10G tank. However, it will most certainly need more room as it grows, so 20G for an adult fancy becomes the absolute bare minimum. You will also need a whole lot of filtration, regardless of tank size--at least 10x gal/hour in relation to the tank's volume. So for a 10G tank you need a filter that runs at at least 100 gal/hour.
(Btw, don't reference the tank in my sig for stocking. Right now I'm cycling another tank so that I can spread out my goldie group, and even with ample filtration on the current tank I'm doing 80% water changes 3 times a week to keep the water nice.)