Combine the vegetables. In a large bowl, add the shredded green cabbage, red cabbage, grated beet, and grated carrot. Toss together well.
Salt and massage. Sprinkle the sea salt over the vegetables. Using clean hands, massage and squeeze the mixture firmly for 5–8 minutes. You're looking for the vegetables to release their liquid and start to soften. Don't rush this step — the brine is what protects your ferment.
Add flavour. Mix in the grated ginger and caraway seeds if using.
Pack the jar. Transfer the mixture into a clean 1-litre mason jar, pressing everything down firmly so the vegetables are submerged in their own liquid. Leave about 2–3 cm of space at the top.
Top up if needed. If the liquid doesn't fully cover the vegetables, mix ¼ tsp salt into ½ cup filtered water and pour in just enough to submerge everything.
Weigh it down. Place a small zip-lock bag filled with brine, or a clean stone, on top of the vegetables to keep them submerged. This prevents mould.
Ferment at room temperature. Cover loosely with a cloth or loosely-fitted lid and leave on your kitchen counter, out of direct sunlight, for 3–5 days. Taste daily from day 3. The longer it sits, the more sour and tangy it becomes.
Seal and refrigerate. Once it tastes pleasantly tangy to you, seal the jar and store in the fridge. It will keep for up to 3 weeks.