Cook and stir beef, potatoes, and onion in a large pot over medium heat until beef is no longer pink. Cover and continue simmering, stirring occasionally, until potatoes are tender, 10 to 15 minutes.

Meanwhile, place carrots and 1/2 cup water in a microwave-safe bowl. Microwave on high until carrots are tender, about 5 minutes.

Drain grease from beef mixture. Add carrots, juice cocktail, green beans, corn, remaining 1 cup water, salt, and pepper to the pot. Simmer over low heat until flavors are well blended and soup is hot, about 30 minutes.