This is an unbelievable way to forget about the real world, turn your phone off and drift from island to island.
A great place to start is in the Peloponnese at a small town called Poros. You can reach Poros from Athens on a short 1 hour fast ferry ride from Piraeus the main port.
There are several charter yacht companies that t operate from here, and you can easily explore islands like Hydra, Spetses, and many more. This part of Greece suffers a lot less from the strong summer winds, you can charter yachts from 32 ft up to 60 ft, and in general a a mid size yacht for 6 adults and up to 2 crew will run around $2000 for a week, you will use almost no fuel, so budget about $50 for this, and port fees to anchor are just a few dollars a night. You can add a captain and stewardess for another $300 per day, if you would rather not do the sailing, cooking or cleaning yourself that's a 7 day yachting holiday for around $700 a person . This is a very affordable version of charter yachting, you are also able to get real close up to beaches, often tying yourself to a tree close up to the beach.
If you have 2-3 weeks then you could venture out to the Cyclades to visit places like Naxos, Ios, and Mykonos.
There ar airfares to Athens from under $1000 fom LAX or SFO. We have great knowledge for yachting vacations in Europe.