I think chess could be classified as a sport. The only thing that might hold it back is how physical it is, but all sports are a combination of the mental and physical. Lets look at it this way. Sprinting is a sport. It may seem that it is entirely physical, but talk to anyone who is top of their field in it and they'll talk about the mental aspect too. At lower levels, not so much. I believe chess is a mirror to that. At lower levels, it appears to be nothing but a mental exercise, but at the top tiers you need to be taking care of your body as well. Eh, it could go either way.
Obviously these are generalizations.
Chess requires absolutely no skill, other than the ability to move the pieces. This is one of the main reasons that it is not viewed as a sport.