Our guest bedroom/laundry drying room is quite large, we put the laundry out on a rack in front of the radiator (which tbh, doesn't come on often) and run a dehumidifier next to it. Despite being open plan, it dries a full load overnight and produces a couple of litres of water from the dehumidifier. It dries without the dehumidifier in summer, of course, but in winter it would sit there soggy for a couple of days unless I cranked up the heating in that room. Never bothered with a tumble drier.
I just dangle my swimming towel from the kitchen door where it dries fine. I can't say I notice the chlorine odour (it's not actually chlorine, it's the organics like chloramines, and they volatilize quite quickly). That said, our pool isn't that chlorine-y. I certainly don't wash it every time, that way lies madness.