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A 6 x 8 foot plot can support a couple tomato plants, maybe some bush beans, and some lettuce.
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A 10 x 18 foot plot can hold all that, plus a couple space-consuming squash plants and cucumbers, and maybe some carrots or beets.
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A 20 x 24 foot plot can hold all that, plus peppers, leeks, broccoli, turnips, and maybe some herbs.
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A 40 x 60 plot allows you more of everything, plus some bigger items, such as corn (corn isn't worth growing unless you can have a dozen or more plants because otherwise they don't pollinate or pollinate completely, and you end up harvesting gap-toothed ears) and asparagus or rhubarb.
Allow for succession planting: If something is harvested early in the summer, lettuce, say, or peas, you can then free up that space for another crop, such as carrots. Succession planting is a good trick, but to pull it off, you may need to do some research as well as some trial and error — and be willing to invest the time and effort.
You could also plan for a constant vegetable harvest by intercropping, or interplanting. This method is simple: Just have two different plants share the same part of the garden in an alternating or checkerboard pattern. This setup can look rather nifty, but it has practical advantages as well. Smaller, faster-maturing plants can grow with larger, slower-growing ones and so you always have something to harvest. And plants that appreciate a little shade can grow in the shelter of taller ones (have pole beans next to lettuce or spinach, for example).