Trim and peel the beets. Halve each beet and cut into roughly equal pieces that are about 1/2" by 1".

Put the beets in a non-reactive 2-quart pot along with the water, vinegar, sugar, and salt and simmer gently, covered, until the beets are very soft but not falling apart, about 2 hours.

Drain the beets of their liquid and allow them to cool. Place the beets on the rack of a dehydrator and dehydrate at 145 degrees F until chewy and raisinlike, 4 to 5 hours.