Place beef into a large container and cover with water. Soak in the refrigerator to remove excess salt, 8 hours to overnight. Drain, rinse, and cut beef into small cubes.
Bring cubed beef, 10 cups water, and peas to a boil in a large pot over high heat. Reduce heat to medium-low, cover, and simmer until peas have softened and mostly dissolved, about 30 minutes. Stir or whisk frequently to help peas dissolve and to keep soup from burning on the bottom of the pot.
Stir in carrots, turnips, and black pepper. Continue simmering until vegetables have softened, about 15 minutes more.