As I recall RG6 is what is used for long runs of TV distribution and is 75 ohms. General concenses is that nothing but 50 ohm is usable for ethernet. This gets argued once in a while. Even using 75 ohm terminators on both ends is not sufficient as they will mess up the voltage level of the collision detection circuit. You could get broadband bridges.
People do run broadband ethernet over video cabling systems.
Probably not worth the cost for what you want.