Combine water, soy sauce, sugar, and sake in a bowl to make broth.
Arrange beef, tofu, Chinese cabbage, yam noodles, shiitake mushrooms, enoki mushrooms, and green onion on separate plates on the table.
Heat oil in an electric skillet or a large skillet set over a hot plate at the table. Add beef slices; cook and stir until browned, about 1 minute. Pour in some broth; bring to a boil. Stir in tofu, cabbage, noodles, shiitake mushrooms, enoki mushrooms, and green onion; simmer until softened, about 5 minutes.
Ladle cooked sukiyaki mixture into serving bowls. Replenish broth in the skillet.
Crack each egg into a small bowl and beat lightly. Serve sukiyaki alongside eggs for dipping.