According to
http://www.how-to-grow.co.uk/how-to-grow-soft-fruits-whitecurrants.htm
http://www.how-to-grow.co.uk/how-to-grow-soft-fruits-gooseberries.htm
http://www.how-to-grow.co.uk/how-to-grow-soft-fruits-blackcurrants.htm
All grow well in containers ? Is this true ? According to Phil Corbett of cool temperate leveller gooseberries do.
The same website says that raspberries like containers which surprises me.
in my personal experience the containers have to be large, topping up with new and fresh compost can be difficult, pruning has to be done to keep the plants from outgrowing the pots and the yields are according to the limited growth - but if pots are the only solution then you can but try it - look for varieties that lend themselves to this king of cultivation as they will always do better. I grow some blueberries and blackcurrants in pots as I ran out of ground space and they are fine
raspberries probably do good as they probably like the deepness that a pot can offer and their roots are quite small
Sadly I have to grow blueberries in containers - my soil is too limey for them.
Bob Flowerdew said that raspberries do not like containers.
All these do best in open ground, but if you do have to grow them in containers, calendula has hit it on the head when she says containers have to be large - the larger and deeper the better. I reckon the whitecurrants and gooseberries may probably get away with a smaller pot, say 30 cm diameter minimum, but the blackcurrants really want as big and deep as you can, and they don't like to be dried out.
Best of luck.