I also looked into a washer dryer unit that I could plug into my sink. They are really common in Europe but not in the United States. The problem is that American voltage is too low to run a dryer without a dedicated outlet.
With the way that batteries are going, you could probably make a dryer that has a battery in it so it doesn't need a dedicated high voltage and high amperage plug. Because all dryers need counterweights, the battery really won't add weight to the dryer.
Without technology changes, you can still get an apartment washer dryer unit. Most appliance store should be able to special order one for you. The problem is that they will take all night to dry your clothing, because you are stuck with the limited voltage and amperage that comes out of the standard American wall outlet.
If your criteria are specifically a single-household-only washer and dryer setup, and not just avoiding poorly-maintained central facilities, then my suggestion will not suffice.