Pour the oil into a heavy-bottomed stockpot or lobster pot. Add all the other ingredients and cook, stirring, over medium-low heat until the vegetables have all softened but not colored, about 15 minutes. Pour in 6 quarts water, raise the heat to high, and bring the water to a boil, then lower the heat and let the liquid simmer for 2 hours.
Carefully strain the stock through a fine-meshed strainer set over a large bowl or pot, pressing down on the solids with a wooden spoon to extract as much liquid and flavor as possible. Let the stock cool, then refrigerate for up to 1 week or freeze in batches to be thawed and used as needed.
Yield: 5 quarts