It can't be 0%, because then cheese wouldn't exist, right? :)
After all, nope. Cheese is not fat only.
There is also protein and water. So if you'd remove all the fat, cheese will still contain water and protein.
Many hard cheeses contain about 30% fat and 30% to 10% protein. Gruyère for example has 32% fat and 28% protein (see here, in German). The softer the cheese the more water it contains (and sometimes also more fat).
In Switzerland we don't know "low-fat", but instead "half-fat", "fourth-fat" and "no-fat". A sour fresh cheese or curd is available as no-fat (less than 1% of fat). Also Schabziger/Sap Sago (world worst cheese according to Cheese Mistress, scroll down to Sap Sago :-) is fat free. Cottage cheese is sold as fourth-fat (6% fat contents). Some hard cheeses, especially Alp cheeses, are half-fat.