Leaving aside whether it's a good idea to use the same space for raspberries immediately again ...
I'm very pleased with my "Joan J" which has lovely large, deep red and juicy fruit. They'd be even better with a little more care than I give them. I bought six plants ten or eleven years ago and they now occupy a bed 14' x 4'. I planted them about 18" apart and used the most errant suckers to finish the row the next year. They rapidly form new clumps out from the row.
Access for maintenance isn't really a problem as all you have to do is cut all the shoots down to ground level over winter. I weed, feed and mulch them then (they haven't had much weeds in among them). You need to decide how you are going to support the canes and get the framework in. I've tried various things and haven't got a satisfactory set up yet.
Access for picking is the issue, so the question is whether your 6' x 6' has clear space around it (grass, path, whatever) or is squeezed between other fruit bushes or such.
I think I'd put in six plants in a double row (2 x 3) at 24" each way.