Prepare an ice-water bath; set aside. Cover beets with cold water by 2 inches in a large saucepan. Bring to a boil; reduce heat. Simmer beets until tender when pierced with the tip of a knife, about 30 minutes. Transfer beets with a slotted spoon to ice-water bath. Discard cooking liquid. Trim beets and rub off skins with paper towels, or peel beets with a paring knife. Cut beets into very thin rounds; transfer to a large bowl.
Bring ginger, vinegar, sugar, and salt to a boil in a medium saucepan, stirring until sugar is dissolved. Pour liquid over beets; stir. Let stand until completely cool, at least 1 hour.Transfer to an airtight container, and refrigerate up to 1 month.