Prepare a jar: Thoroughly wash and dry a wide-mouth quart mason jar in hot, soapy water.
Prepare the cabbage and carrots: Cut the cabbage into quarters and cut out the core. Slice the cabbage as thinly as possible, working from crown to stem end. Place shredded cabbage into a large nonreactive bowl. Grate your carrots or other vegetables with a box grater, and add to the cabbage. Add the salt and mustard. Using your hands, toss to combine all ingredients, crushing the mixture until it begins to release liquid, about 5 minutes.
Pack the sauerkraut: Place a wide-mouth funnel in the mouth of the jar. Pack the mixture into the jar, pressing down to squeeze out as much air and release more liquid. Tightly packed, there should be just enough to fill the quart jar to the shoulder, and liquid should rise above the cabbage mixture. Fill a smaller jar, like a half-pint or four-ounce jar, with water and seal. Place on top of the cabbage mixture to keep it submerged under the juices.
Ferment the sauerkraut: Cover the jar with a clean towel, and allow to stand at room temperature. Check daily to make sure the cabbage is still submerged in liquid. If it is not, give the mixture a good press to release more juices. If the cabbage still does not stay submerged, add just enough water to bring the level up. After two or three days you should see bubbles forming. Eventually you may see scum or even mold on top. This is normal; just skim it away. Begin tasting your kraut after the third day. When it has reached the desired level of tartness, remove the smaller jar, screw on a lid, and keep the kraut several weeks in your refrigerator.