If you can get a copy of the superb book, "Square Foot Gardening" by Mel Bartholomew, perhaps from the library, he covers this topic extremely well. There's lots on this site too about square foot gardening if you look.
Depending on the time of year, some of the easiest vegetables for close space gardening include baby beet, spring onions, Chinese greens like Mizuna and Mibuna, radishes, baby Japanese turnips like Tokyo Cross and so on. These all grow quickly and give a good harvest in a small space. Carrots too can be sown repeatedly if you choose early varieties and then later ones and sow accordingly, harvesting them while small.
A wonderful herb not grown that much is Winter Savory, a bit like thyme, which you can pick almost all year round, as well as parsley (not the curly type which doesn't taste as good), chives (which come back each year), and in the summer, basil and coriander, both of which are easy annuals.
I do recommend the Square Foot Gardening book for anyone with a small garden though.

I think Bertie has just about covered it. When you are sowing the seed you can actually sow closer than the packet recommends and just harvest when the plant is slightly smaller. I would add salad leaves the kind you just keep picking the outer leaves of rather than lettuce.