In a heavy skillet, cook onion in hot oil over medium low heat. Stir in ketchup, celery, brown sugar, Worcestershire sauce, vinegar, mustard, water, and salt. Cover and simmer for 15 to 20 minutes.
Lightly score hot dogs diagonally at 1-inch intervals. Add hot dogs to sauce; cover and simmer for about 15 minutes longer. Toast split buns. Serve hot dogs over toasted buns.
Barbecued hot dogs serves 4 to 6.