Make the beet pickling juice by placing the vinegar, water, sugar and salt in a medium sized saucepan. Turn the heat on high, until the solids dissolve, then add the beet. Lower the heat and simmer for 30 minutes, or until the beet is tender and cooked. Remove the beet and save for another use. Let the brine cool. Place the peeled hard-boiled eggs in a large bowl, then pour enough of the brine to cover the eggs. Then cover and refrigerate overnight (8 hours) up to 2 days (48 hours).