I don’t think Fox News even claims to be fair and balanced. I think their view is that since all other channels are unbalanced to the left, it is their aim and duty to be unbalanced to the right.
While accepting that there is not such thing as total lack of bias, and accepting that there exist channels biased to the left, I think their claim that there are no channels to the right is false and their conscious lean to the far right is appalling. But it is a money driven business and their customers get what they want.