There are careers after a fine art degree both as a degree for graduate jobs and in the field of art such as a teacher, artist obviously and art administration such as in a gallery, local authority or museum.
But it is going to be hard to get a good job with an fine arts degree because most people think its just 3 years of messing about with paint and drawing naked people.
My wife did a degree in fine art but never worked in the arts, she worked in shops and in engineering.