Weigh the cored, sliced cabbage and grated beets together. They should land around 1.2 kg / 2.6 lb total, which tells you exactly how much salt you need for a clean 2% ferment.
Combine the cabbage and beets in a large mixing bowl. Sprinkle the fine sea salt evenly over the top.
Massage the salted vegetables with your hands for 8 to 10 minutes, squeezing and folding, until they go limp and pool in their own liquid at the bottom of the bowl.
Stir in the caraway seeds and sliced garlic.
Pack the mixture tightly into a 1-quart / 1 L fermentation jar, pressing down with your fist or a wooden tamper after each handful to force out air bubbles.
Pour any leftover brine from the bowl over the top. The liquid should cover the vegetables by at least 2.5 cm / 1 inch; top up with filtered water and a pinch of salt if it falls short.
Weigh the vegetables down with a fermentation weight or a folded outer cabbage leaf so nothing floats above the brine. Seal with an airlock lid or a loosely fitted regular lid.
Set the jar on a plate, since it may bubble over, in a spot away from direct sun, at 18 to 22 C / 64 to 72 F, for 5 to 7 days. Press the vegetables down once a day to release gas and keep them submerged.
Start tasting on day 5. When it's tangy, a little fizzy, and the beets have turned everything a deep pink, move the jar to the fridge.
Serve cold, straight from the jar, using a clean fork every time to keep it from spoiling early.