Before I got my plot I grew in 2 beds, roughly 12 sq yd each, in my garden. I used to put stuff in much closer than recommended without any real issues - plants were sometimes smaller but I looked on 2 small cabbages as being better than 1 large one and it can be a bit more difficult to get a hoe in to weed so more care is needed to avoid taking out a veg plant. Rather than planting in a straight line I also used to offset every second plant in a zig-zag pattern which creates a bit more space for each. One thing to seriously consider is the space (and time for other than earlies) taken up by potatoes. I used to grow 1 row (14ft) of 1st earlies which meant that the ground was freed up for a second crop sometime in July depending on when the seed spuds went in. Another is onions, which are cheap to buy and are one of the few crops I find that don't have vastly superior 'home grown' taste - if you do grow them I'd do it in clumps of 3 or 4 (again they'll be smaller than single planting but you'll get more bulbs in the allotted space). I'd also look at growing your carrots in containers if possible. They will, almost certainly, need protecting from carrot fly and covering them in the ground will also add to the difficulty of maintenance. Is that PSB above the parsnips? Again that's another crop that will take up a relatively large amount of space for a long period but is something that I did grow in the garden as we like it and it provides something when there's not much else around to harvest.
Remember that it's much easier to thin things out if you do overplant than it is to fill in the gaps if you underplant!
I'm sure this doesn't need saying, but especially given your limited space, try to only grow stuff that not only do you and your family like to eat but that you will actually use (my plot neighbour likes cabbage but he certainly doesn't get through the 60 or so he grows every year!) so that way you get maximum benefit from the space. it might be worth having a look for square foot gardening to get some idea about how tightly things can be packed in. And finally, give it a go and see what happens!
Ps not sure what that is above the swedes(or to the right), sweet potatoes? If so I'd forget them and plan to grow something else.