DS is 5 mo too and I think this is a hard time for everyone. I'm generally pretty happy with everything but I still enjoy my little bits of time to myself and get frustrated when I really wanted to get something done but just fail, or accept that its just an impossible dream.
DS doesn't do anything on his own for more than 10 mins, or 20 with me singing to him (constantly!) He doesn't even sleep during the day unless he's on me, in the sling or in the (moving) pushchair.
I think I cope because I:
a) don't care too much about the houseweork - me and DH get enough done between us to keep us moving. You might not manage that though!
b) plan around DS - he's better in the mornings to I try to do stuff then like throwing together a casserole which I can just switch on in the afternoon when I can't put him down
c) try to go somewhere every day - I walk everywhere so DS will always get a nap either in the sling or pushchair, and its not so bad to be holding him all the time when I'm doing something like the shopping or chatting to mums at a baby group. He doesn't feed well when we're out but I try to make up for it in the afternnon by snuggling up in a quiet room with a book or mumsnet and getting 2-3 hours of feeding/sleeping in (he doesn't do this every day)
Also, I find the staff at our babygroups are more than happy to hold babies for 10 mins to give you a break!
the sling is a lifesaver and when DS was even worse we lived on casseroles made from raw things thrown in a dish and cooked in stock in the oven cos I could safely do that!
I also agree that health implications aside, you'll find weaning just one more thing you don't need to do.
I'm sure its normal to feel like this, but I'm sure you can do practical things to make things a bit better, and if nothing helps, please go and see the GP.
Good luck!