Cook beef shank in a large soup pot over high heat until browned, about 3 minutes per side. Add water, onion, carrots, celery, and bay leaf; bring to a simmer and cook until meat is tender and falling off the bone, about 4 hours. Strain broth and discard solids.

Combine beef broth, beets, and cabbage in the same pot; cook, stirring occasionally, until beets are tender, about 30 minutes. Reduce heat to low; add vinegar, salt, and black pepper.