If you're looking for a place where you can properly unplug and unwind, then I can highly recommend Panormos Villa. Quite literally at the end of the road – about as far as it's possible to get from the island's main town – the villa sits in splendid isolation, perched on a hill overlooking olive groves, orchards and the spectacular beauty of Panormos Bay and the Small Cyclades islands.
Panormos Beach is small, sheltered and (in late June at least), practically deserted, with a beach bar serving excellent salads and souvlaki (and a surprisingly good mojito) among other things. I suppose you could walk to the beach, but it would be a long, hot walk back. And since you'll need a car to get to the villa in the first place, it's far quicker and easier to drive.
In the evening, you can hear nothing but crickets and goat bells from the hills all around, and the view from the terrace as the sun goes down is nothing short of magical.
Bear in mind, this is a relatively isolated place. The water from the taps is not suitable for drinking or cooking, so you need to bring lots of bottled water. Apart from the beach bar, the nearest place to get food is in Moutsounas, about half an hour's drive away, where there are a few bars and restaurants and a small (but perfectly good) convenience store. The kitchen is functional but basic and there aren't really facilities for outdoor cooking. So if you're staying for more than a few days, I'd recommend stocking up on supplies in one of the supermarkets in Naxos town before you head across the island. That said, you could easily get by just on Antonis and Georgina's delicious Greek salads from the beach bar.
Ioannis is an excellent host – helpful and communicative, quick to respond to questions and knowledgeable about the area.
We had a wonderful, peaceful and relaxing week at Panormos Villa and have no hesitation recommending it as a perfect place to get away from the stress of everyday life.