Cook the beets (boil or roast) until nearly tender. Total timing varies, but is usually 20-30 minutes, depending on the size and age of the beets.
Cool slightly, then slip the peels off the beets, trim off the stem and root ends.
Cut the beets in half and slice into 1/4-inch slices. Small beets under 1 1/2 inches in diameter can be left whole.
Combine the vinegar, water, sugar, and salt in a nonreactive saucepan and bring to a boil, stirring until dissolved, then remove from the heat.
Pack the prepared beets into clean, hot jars, dividing the dill weed and dill seed among the jars. Leave about 1 inch of headspace above the solids, so you can ensure everything is well covered by the brine.
Pour the hot brine over the beets, covering them and leaving 1/2 inch final headspace. Remove air bubbles, wipe rims, apply lids, and tighten bands to fingertip tight.
For waterbath canning, process pint jars in a boiling-water canner for 30 minutes (adjusting for altitude, see notes). Turn off the heat, remove the canner lid, and let jars rest 5 minutes before removing. Cool 24 hours, check seals, and store sealed jars in a cool, dark place for up to 1 year.
For refrigerator pickles, cool, cover, and refrigerate.