Have you looked into the small washers? They're sometimes recommended for use in RVs. They're a step up from hand-washing, but take about the room of a 5 gallon bottle of water. Paired with a hose attachment for your sink or a hand-held shower head, it's plenty for washing flats and covers. And a drying rack for drying - put it near the heat source and they should dry over night. And if you're essentially hand-washing them, you can do it in your time (5 minutes washing, go deal with the baby, back to washing, go answer the phone, back to washing, check on dinner in the oven, etc.). Rinse, agitate, dump. Soap, agitate, dump. Rinse, agitate, dump. Repeat rinse at least once. Done. It doesn't take nearly the amount of time a machine does. You'd probably have to be doing it every day at first though.
If that doesn't work for you, I'd probably explore a service. Sitting in the laundry room/laundromat waiting for the laundry is not a good use of my time. There are so many things I'd rather be doing.
But as for your concern about the poops and other people - so long as the babe is EBF, the poo washes clean away. It's completely water soluble. Formula poo is pastier and doesn't tend to wash as clean. And once they start solids, all bets are off. At that point I recommend another approach - liners, diaper sprayer, scraping, whatever works for you. But the poo can't go in the washer once they start solids, it clogs up the machine.
Or, you could always do a combo of all of the above.
Oh, and something else you might want to explore is here we have a diaper service that composts disposables. You have to use a particular brand, but rather than putting them in a landfill, they get picked up and composted. I think cost-wise it's on par with a regular service, although I haven't really explored it much. Maybe look if you have something similar near you.