Place beets, onion, and garlic in a pot. Pour the water and vinegar over, and add the salt and sugar. Cover and cook for 60 minutes until the beets soften and are easily pierced with a knife.

While the beets cook, bring another pot of water to boil. Add the eggs, and cook for 11 minutes. Transfer to a bowl of ice water to stop the cooking. Peel and set aside.

Once the beets are tender, use a slotted spoon to remove the vegetables and let them cool in a colander. Let the cooking liquid cool to room temperature. 

Arrange the jar: Gently place the eggs in the jar, and cover with the cooking liquid. Make sure the jar is full, but not packed. Place 2-3 pieces of beet on top to help keep the eggs submerged. Close the jar tightly, and let sit at room temperature out of direct sunlight for 12 hours. Transfer to the refrigerator for 5 days.

Every 24 hours, gently shake the jar to make sure the eggs are fully coated in the liquid. The eggs will keep for up to 2 weeks refrigerated.