Many people do raspberries and blackberries in raised beds all the time with great success. They'll grow great in Mel's Mix, just make sure there isn't a hard bottom on your box.

Blueberries need a pH in the range of 4.5-5.5, but Mel's Mix has a pH of around 7. Instead of 1/3 peat moss, make it 1/2 or more to help lower the pH even more. The hardest part with blueberries is lowering the pH, but once you get it down, growing them is a piece of cake. Just top dress with a small handful of sulfur each spring to keep the pH down and fertilize with Miracle Grow for acidic loving plants.

Now, hopefully my explanation isn't too confusing, but google "growing blueberries at home" if you need more information. It's pretty simple once you take care of the pH.

The thing about blueberries is your water can raise the pH as well. So you must be careful.

Nonetheless, if the OP has, say, a 4x4 box, a nice 6' trellis on two sides for the blackberries might work, and depending upon the variety it may in fact stay in that space without too much effort. I'm not sure the return on a 4x4 for raspberries is that high. I used to have a 25' row and that was very fine, and I can't say I would have maintained a 4' row. In my view take the rasps out of the box and make a nice long row (or two) in the yard and work the soil well and put up quality posts and wire. Might as well put blackberries in the yard too and make maybe an 8' wide trellis and just dedicate the space to ease the work.

Dan