Cut off all but 1 inch of beet stems and roots; wash. Do not peel.
In a 6 to 8 quart Dutch oven combine beets, onions, and salt. Add water. Bring to boiling; reduce heat to medium-low. Cover and simmer for 20 minutes. Cool slightly. Slip skins off beets; discard. Cut beets into large pieces.
Stir in sugar and lemon juice. Return to boiling; reduce heat. Simmer, uncovered, for 10 minutes. Cover and chill, if desired.
To serve, ladle soup into bowls, using one onion half per serving. Sprinkle cabbage over soup. Makes 8 side-dish servings.