Clothes recyclers are common appliances on starships, and are basically combined washing/drying machines. The actual process used to clean clothes is more of a dry-clean; it uses some combination of reusable chemicals and sonic vibrations to clean clothing, almost like an ultrasonic cleaner. Some species are able to detect a gentle humm from the machine when it is running, but it is not aggrivating to most.
It is common for clothes recyclers to be part of a person's quarters, often set into the wall so it doesn't take up floor space. It operates by tossing clothes in the top; the clothes are dropped out the bottom when its done. The idea behind the clothes recycler is to toss the garments into the machine as soon as you disrobe, and then they will be clean the next day to wear again or put away as desired.
As such, clothes recyclers in quarters are usually pretty small; just enough for maybe two to four outfits worth of clothes. On the legacy, the recyclers are large enough to handle the sheets; but on some ships, there is a separate laundry room for washing sheets in a larger recycler.
They are very energy efficient, low maintenance, and re-use their own fluids for quite a bit, so it makes sense to have them wherever they are needed rather than to have separate laundry facilities. As such, the Legacy's common showers and the gym would have recyclers as well for the towels, and there are probably some in the kitchens as well.