This is a commonly used expression, particularly among the older generation. It is used in an informal setting and although it may appear to be it is not in fact an insult.
This expression is used as another way of saying “no chance.” It is often used in situations when someone is pushing their luck and asking for more than is necessary and thus when asking for something else they are told “fat chance.” Interestingly the English phrase “slim chance” has a similar meaning with only a slightly different undertone despite sounding as though it should mean the opposite.
The expression has no definite origin or first usage. It is suggested that the phrase “slim chance” was used first in the very literal sense of meaning “little or no chance.” However it was felt that a phrase was needed to be used in a more sarcastic context and thus the opposite of “slim” was employed.
An example of the usage of this expression can be found in the title of an article from The Science Bit which reads “chocolate keeps you thin? Fat chance!” This is an article written in response to an article published on the BBC news website which claimed that eating chocolate could help people stay slim. The article completely refutes all arguments made that chocolate could help people manage their weight and therefore the term “fat chance” is used in order to create a sarcastic tone.