Put the water and ground beef in a Dutch oven or soup pot over high heat and bring to a boil. Peel and chop the onion while waiting for this to boil. Add the onion, reduce the heat to medium, and stir to break up the meat a bit.
Add the tomato paste, ketchup, chili powder, Worcestershire, vinegar, salt and pepper and stir until the tomato paste is dissolved and everything is mixed well, breaking up the meat chunks as you go.
Simmer over low heat about 15 minutes, until very thick, stirring now and then.
Serve on hot dogs with grated Cheddar cheese if you want chili cheese dogs. They didn't suggest that so maybe it's not a Carolina thing, but around here you get chili cheese dogs if you get chili on hot dogs. ;)