If only large beets are available, quarter or chop them into chunks after peeling in Step 1.
Wash beets, leaving on the roots and 1 inch of tops. Place in a 4- to 6-quart Dutch oven or kettle; add water to cover. Bring to boiling; reduce heat. Simmer, uncovered, until tender, about 25 minutes. Drain and discard cooking liquid. Cool beets slightly; trim off roots and stems. Slip off the skins.
In the same Dutch oven or kettle combine vinegar, water, and sugar. Place allspice, cloves, and cinnamon in the center of a 7-inch square of 100-percent-cotton cheesecloth; tie into a bag and place in Dutch oven. Heat to boiling; reduce heat. Simmer, uncovered, for 5 minutes.
Pack beets in hot, sterilized half-pint canning jars, leaving 1/2-inch headspace. Carefully add the boiling pickling liquid, leaving 1/2-inch headspace. Discard the spice bag.
Wipe jar rims and adjust lids. Process in boiling-water canner for 30 minutes. Remove jars from canner; cool on racks.