We basically followed the recipe below. Used the big FYIPA-sized mason jar. It didn't really work to keep something on the cabbage to keep it submerged though. We did add caraway seeds, and cut it off after about 3 days.
1 medium head green cabbage (about 3 pounds)
1 1/2 tablespoons kosher salt
1 tablespoons caraway seeds (optional, for flavor)
2-quart wide-mouth canning jar (or two-quart mason jars)
Canning funnel (optional)
Smaller jelly jar that fits inside the larger mason jar
Clean stones, marbles, or other weights for weighing the jelly jar
Cloth for covering the jar
Rubber band or twine for securing the cloth
Clean everything: When fermenting anything, it's best to give the good, beneficial bacteria every chance of succeeding by starting off with as clean an environment as possible. Make sure your mason jar and jelly jar are washed and rinsed of all soap residue. You'll be using your hands to massage the salt into the cabbage, so give those a good wash, too.
Slice the cabbage: Discard the wilted, limp outer leaves of the cabbage. Cut the cabbage into quarters and trim out the core. Slice each quarter down its length, making 8 wedges. Slice each wedge crosswise into very thin ribbons.
Combine the cabbage and salt: Transfer the cabbage to a big mixing bowl and sprinkle the salt over top. Begin working the salt into the cabbage by massaging and squeezing the cabbage with your hands. At first it might not seem like enough salt, but gradually the cabbage will become watery and limp — more like coleslaw than raw cabbage. This will take 5 to 10 minutes. If you'd like to flavor your sauerkraut with caraway seeds, mix them in now.
Pack the cabbage into the jar: Grab handfuls of the cabbage and pack them into the canning jar. If you have a canning funnel, this will make the job easier. Every so often, tamp down the cabbage in the jar with your fist. Pour any liquid released by the cabbage while you were massaging it into the jar. Optional: Place one of the larger outer leaves of the cabbage over the surface of the sliced cabbage. This will help keep the cabbage submerged in its liquid.
Weigh the cabbage down: Once all the cabbage is packed into the mason jar, slip the smaller jelly jar into the mouth of the jar and weigh it down with clean stones or marbles. This will help keep the cabbage weighed down, and eventually, submerged beneath its liquid.
Cover the jar: Cover the mouth of the mason jar with a cloth and secure it with a rubber band or twine. This allows air to flow in and out of the jar, but prevents dust or insects from getting into the jar.
Press the cabbage every few hours: Over the next 24 hours, press down on the cabbage every so often with the jelly jar. As the cabbage releases its liquid, it will become more limp and compact and the liquid will rise over the top of the cabbage.
Add extra liquid, if needed: If after 24 hours, the liquid has not risen above the cabbage, dissolve 1 teaspoon of salt in 1 cup of water and add enough to submerge the cabbage.
Ferment the cabbage for 3 to 10 days: As it's fermenting, keep the sauerkraut away from direct sunlight and at a cool room temperature — ideally 65°F to 75°F. Check it daily and press it down if the cabbage is floating above the liquid.
Because this is a small batch of sauerkraut, it will ferment more quickly than larger batches. Start tasting it after 3 days — when the sauerkraut tastes good to you, remove the weight, screw on the cap, and refrigerate. You can also allow the sauerkraut to continue fermenting for 10 days or even longer. There's no hard-and-fast rule for when the sauerkraut is "done" — go by how it tastes.
While it's fermenting, you may see bubbles coming through the cabbage, foam on the top, or white scum. These are all signs of a healthy, happy fermentation process. The scum can be skimmed off the top either during fermentation or before refrigerating. If you see any mold, skim it off immediately and make sure your cabbage is fully submerged; don't eat moldy parts close to the surface, but the rest of the sauerkraut is fine.
Store sauerkraut for several months: This sauerkraut is a fermented product so it will keep for at least two months and often longer if kept refrigerated. As long as it still tastes and smells good to eat, it will be. If you like, you can transfer the sauerkraut to a smaller container for longer storage.
Sauerkraut with other cabbages: Red cabbage, napa cabbage, and other cabbages all make great sauerkraut. Make individual batches or mix them up for a multi-colored sauerkraut!
Canning sauerkraut: You can process sauerkraut for longer storage outside of refrigeration, but the canning process will kill the good bacterias produced by the fermentation process. See this tutorial from the National Center for Home Food Preservation for canning instructions.
Larger or smaller batches: To make larger or smaller batches of sauerkraut, keep same ratio of cabbage to salt and adjust the size of the container. Smaller batches will ferment more quickly and larger batches will take longer.
Hot and cold temperatures: Do everything you can to store sauerkraut at a cool room temperature. At high temperatures, the sauerkraut can sometimes become unappetizingly mushy or go bad. Low temperatures (above freezing) are fine, but fermentation will proceed more slowly.