To make sauerkraut in a mason jar, sprinkle salt over thin cabbage slices, and transfer the cabbage into a clean jar while pouring out fluids from the cabbage. Insert a small jelly jar into the opening, weigh it down by adding clean stones, and cover the mouth with a cloth tied with a rubber band. Push the cabbage down every few hours, add salted water if necessary, and ferment for three to 10 days.
Cut 1 medium-head, green cabbage into quarters with the core trimmed, slice the quarters into eight wedges, cut further into thin ribbons, and remove any wilted outer leaves. Put the cabbage slices in a large mixing bowl, add 1 1/2 tablespoons kosher salt, and massage the cabbage gently for up to 10 minutes. Add 1 tablespoon caraway seeds if desired.
Use a canning funnel to transfer the cabbage to a jar easily. After weighing the cabbage down with clean stones and covering it with a cloth, press it occasionally within 24 hours to release its liquid. Combine 1 cup water and 1 teaspoon salt, and pour some into the jar if liquid does not form after 24 hours.
Once fermented, store in a cool area away from direct sunlight. Press the cabbage in the next days if cabbage floats. Taste after three days, remove the stones, secure with its cap, and store in the fridge for two months or more.