You could do web design at home through freelance work and work at home as a freelance web designer without getting a degree in web design or having any other degrees as long as you're good at doing web design and can make a nice looking professional website design for others.
You need to have experience as a graphic designer and know some website coding and programming as well.
To be hired as a full time web designer you'll need to have a Bachelors degree in computer science, business or communications and complete college programs in Graphics Arts and Web Design.