laundry facilities on Crete or Naxos?

Of course, there are ways of doing laundry in all the places you mention, including hand washing or hotels that do your laundry so I think you are not asking the general question ways of doing laundry but are looking for a laundromat or a drop-off service. If I am wrong and this just a general survey of whether people in Naxos and Chania wash their clothes, read no further. If the question is about laundromats or drop-off laundries, in Naxos, there is, to name one place, Laundry Clean Express. Just Google those words and Naxos for the address or just search for "laundromat Naxos". In Chania, there are at least three laundromats and doing a Google search with "laundry Chania Greece" will find them for you.

Yes so ask your hotelier.

We sometimes do laundry at laundromats (family of five, so doing a few loads on a multi-week trip is needed). We used a laundromat in Chania (Crete). Easy peasy. Google-mapped "laundromat" and went to the one closest to us. Several drop-off options popped up as well (leave you r laundry and pick up later)

In a pinch you can do it in your room, either bathroom or kitchen sink.

If you have a balcony most accommodations will have a drying rack with clothes pin.

Some places will have laundry service, especially Airbnb accommodations that sometimes include a washing machine.

One of the issues of travel is where do you do your laundry on a holiday!

We always purchase laundry detergent from a store when we arrive on our first island. My wife washes her clothes in the sink and I take mine in the shower and use the detergent and shower water to stomp them clean. The balcony rack that virtually every small Greek hotel provides plus the sun works amazingly well. No must no fuss no bother. Works every time.

I pack a sandwich baggie of powered detergent from home. If I don't have a washing machine available,, a tablespoon of the powdered detergent in the sink gets the stink out of my clothes. Hang to dry as the locals do.

Stanbr, his wife and me are like minded thinkers when we travel to Greece.

Nothing wrong with doing laundry in your room's sink!