4'x6' is not very much space - only 24 square feet. You will need to reduce your list significantly. Forget any of the cucurbits (squashes/cucumbers) or melons unless you are able to rig up some sort of vertical trellising. These become rather good sized vines and a single plant can eat up your entire chunk of real estate :-)

Forget the asparagus also. This is a perennial veggie (returns year after year) and you need a bunch to make a reasonable harvest and you need to allow about 3-5 years before you start harvesting. Again, the number of plants you would need to make growing it worth your while for a single serving would take up your entire plot.

Keep your choices down to the smaller veggies and those that you can do succession plantings with - any of the greens, peas and beans, radishes, carrots. Peppers and tomatoes should do well for you but both these plants can get big, so plan accordingly. You may only have room for one or two of each at the most.

Corn is just not practical in a small space - grows too big for too little harvest. Perennial herbs like the rosemary and chives often like it drier and leaner than other veggies, so sometimes planting in a container is more suitable

gardengal speaks from experience! I'll add that you can get a lot of milage if that small spot if you investigate the many different kinds of 'greens' for an extended harvest period. If you like arugula, I'll bet that you can add to that list.

It might be worth trying vining cucumbers, yellow summer squash, and zucchini on sturdy tomato-type cages. I'm not so sure that the other cucurbits would work unless it was really really sturdy. Snap peas would work well with such a cage, as well as other vining beans. LOTS of youtube videos available on different techniques from which you could obtain some inspiration.

You could also look at your list a different way.

Some crops are early season - such as peas and some lettuce and are best followed with varieties that will suit the summer warmth, or be followed by something that enjoys the heat (beans, for example).
Some are very quick to come to harvest - radishes and micro-greens. You can plant them many times over the length of the growing season - well into autumn, too. Just a few at a time, or a little patch.
Some sit in the bed for most of the growing season before they start to produce - tomatoes, onions and corn. You'll have to plant around them, take advantage of the shade they cast.

As it's your first season with the new plot - quick crops might be useful for you because you will be able to clear the bed, or part of it, several times over the season and be able to feed the soil with compost to build your soil fertility and discourage pests each time you clear.

You'll also be able to see which varieties taste best from that plot. Some will be delicious. Others average. And others will leave you wondering how anything could taste so bad.

If there's some way you could rig up a 'towel rail' along the side or end of your bed you could add hanging baskets for items such as cocktail tomatoes, one-season herbs, asparagus peas, and so on. Make sure the rail is very sturdy, though. Wet potting mix can be extremely heavy.

Allocate some space at home for raising your own seedlings.

If you want to start beans in pots do so in pots that are at least three inches deep - and use fresh bean seed to minimise disappointment. Beets will transplant. Carrots less likely. Peas yes - and it's fiddly, particularly if you used grow-cells. Planting straight into the ground is 'better'. Lettuces definitely can be raised in cells or small pots. Keep them moving on into bigger pots until you have room to plant out in your plot. Once those roots start circling the pot they're living in, it's a signal for flowering to start and leaf-growing to stop.

You might want to acquire a water-proof shallow tray for transporting transplants in pots from home to you plot, too. Saves having to clear up spills in your vehicle.