Roughly chop all of the vegetables into 2-inch pieces; precision is not important.

Place all of the ingredients in a large stockpot and set over high heat.

Bring to a boil, then reduce heat to maintain a slow simmer. Do not let the stock boil.

Let the stock cook gently, uncovered, for 6 to 8 hours. Check frequently and skim off any impurities that come to the surface.

Remove from the heat and strain twice through a fine-mesh strainer.

Place strained stock in a clean sauce pot and return to high heat to bring the stock to a boil. Turn the heat down to maintain a light boil; reduce to 2 quarts.

Strain again.

Use immediately, or refrigerate for up to five days.