Answer: You probably can - there are proponents of square foot and intensive gardening methods that would say "absolutely!" If you plan the garden out carefully on graph paper, you'll have your answer. Some of your crops can be grown in succession: for instance, since peas don't thrive in summer heat, you can follow them with a quick summer crop like cukes or squash, and grow them up the pea trellis. Growing vertically is a great way to expand your space, so put fence or trellis up on the north and west sides of your garden (a shorter one on the west side so it won't create too much afternoon shade). If you grow pole snap beans instead of bush, you can use the trellis for them, too, but it'll need to be at least 7' tall. But, since bush beans mature more quickly, you can follow them with another crop, too. Kale, radishes and broccoli are cool-season crops, so plant them in spring and again for a fall crop after the summer squash and cukes are spent. Dense plantings will compete more, so enrich the soil with compost +/or aged manure between plantings, and make sure everything gets plenty of moisture. Thick organic mulch of newspaper covered with straw will help maintain moisture in the soil. Spritz crops with liquid kelp/seaweed 2-3 times during the growing season.
Your vining crops are going to be hardest to keep within the bounds of your plot, so here are some compact varieties to try: 'Sweet Bush Hybrid' cantaloupe, 'Heatwave' and 'Celebrity' Tomatoes, 'Saffron Prolific' yellow squash, 'Sugar Bon' and 'Super Snappy' sugar snap peas, 'Salad Bush Hybrid' or 'Bush Champion' cuke. Baby carrots of any kind will do well, too. Have a good time!
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