Combine your diced beets and red onion with the water in a medium pan, bring to a boil and then simmer for 20 minutes.
Pour your apple cider vinegar, sugar, and salt into the beet pan and stir in.
Rinse and place your peeled eggs into the half-gallon jar.
Place a jar-sized strainer over your jar, and pour the beet pan contents into the jar. (Or strain your beet liquid into another vessel that you can fit your strainer over and then pour the liquid into your jar with the eggs.)
I had enough brine to cover both the eggs and beets & onions in another jar, so this recipe made a bonus jar of pickles for me.
Lid your jar(s), protecting the metal with parchment paper. Whenever I jar something with vinegar or alcohol I put a square of parchment between the liquid and metal jar lids, or I use reusable plastic lids.
Allow the beets to marinate for 5 days in your refrigerator. Shake them daily to make sure that all parts of the egg is getting plenty of beet brine and there aren't any white spots left.Enjoy within a month or so!