Season the beef chunks with salt and pepper as desired.
Put 1-2 cups of flour in a plastic food storage bag. Add the beef chunks to the bag of flour. Twist the bag closed and shake to coat beef with flour. This may need to be done in batches.
Heat a 4-6 quart pot or dutch oven on the stove over medium heat. When hot, add 3 tablespoons oil to the pot. Working in batches if necessary, brown the beef cubes in the oil. Turn the pieces over as necessary to brown on multiple sides.
Remove the browned beef from the pot and set aside on a plate.
The pot should have some nicely browned flour and oil in the bottom. Add a little more oil, if necessary, to saute the vegetables.
Continuing with a medium heat, add the vegetables to the pot and saute until the onions begin to soften. This will take about 20 minutes. Stir occasionally to ensure even cooking and to keep the vegetables from sticking to the bottom of the pot. Sauteing the vegetables allows the flavors to deepen. Do not burn!
When the onions have softened and the other vegetables are a little past crisp-tender, add the beef back to the pot with the vegetables. Add the beef stock and stir to combine all ingredients; bring to a simmer and cook for 2 hours over medium to medium-low heat.
Just before serving, blend 2 tablespoons flour with 1/2 cup cool water until smooth. Bring the stew to a low boil. Slowly add the flour/water mixture to the stew, stirring gently to incorporate. Cook for a few minutes until thickened.