Toss the beets, ginger, orange peel and fine sea salt together in a large mixing bowl. Stir the ingredients well, and allow them to sit for 20 minutes so that the salt brings out the beet juice.
Pack the beets tightly into a 1-quart jar, making sure there's no air pockets. Weigh down the beets, so that the brine covers the vegetables and beets. Seal the jar with an airlock or a very tight lid.
Allow the beets to ferment at room temperature at least 2 weeks and up to 1 month, or until pleasantly sour. If you're not using an airlock, burp the jar every 4 to 5 days. After the beets are sour enough for your liking, transfer them to the fridge and consume them within 6 months.