Place cabbage in a colander over a plate; sprinkle with 1/3 cup salt and toss. Let stand for 30 minutes. Rinse and drain well. In a very large container, combine the cabbage, cucumber, radishes, onions, garlic and ginger.
In a large bowl, combine the water, vinegar, chili paste and remaining salt; pour over vegetable mixture. Cover and refrigerate for at least 2 days before serving, stirring occasionally. May be transferred to small airtight containers and stored in the refrigerator for up to 3 weeks. Serve with a slotted spoon.