The big bang is an extrapolated theory with possibly one piece of evidence: the cosmic microwave background. The CMB may be interpreted in other ways. Not believing in the Big Bang is far removed from being a young-Earth creationist. Many more things have to be wrong (perhaps logic itself) for young-Earthism to be proven right. The field of cosmology is still pretty young and the theories supporting it are much more likely to be changed. But the general idea that the Earth (and the universe) is old (billions of years at least) has many different and unrelated pieces of evidence supporting it.