I think you could cd without a machine but it would be hard. If you already do laundry at a laudramat just buy enough dipes to last as long as your normal intervals (twice a week, once a week, etc). I have a washer but wait up to a week for dipes until I have a full load to save water. The pail reeks by wash day but hey, I don't mind. Pails reek anyway.
Another easier option is to check out the prices of diaper services in your area. Esp. if you live in a place where you have to haul laundry up and down stairs. Good luck!
7yo: "Mom,I know which man is on a quarter and which on is on a nickel. They both have ponytails, but one man has a collar and the other man is naked. The naked man was our first president."