I definitely recommend Loukas and Stathoula's apartments! I travelled with my boyfriend during off season and after spending the first two nights in a mixed dorm hostel room this truly a luxury; fantastic value and I wish we had spent our whole stay in Santorini here.
The apartments are a bit tricky to find at first, as they are 15-20min walk from the main town of Fira but they are less than a 5min walk from the bus stop in Karterados which most bus routes passby.
Despite this it is close to everything you need if you don't mind walking; there is a supermarket nearby, about 5-10min walk, and as there is a fully equipped kitchen in the apartment if you enjoy saving your money to cook your own meals this place is perfect!
We also decided to rent an ATV from the place just across the street (after the bus stop) on our last day, and the gentleman there was very friendly and helpful! Driving it across the island to Oia was definitely a highlight of our trip.
Another plus is that there is a private pool area with lounge chairs for the apartments and although we visited in the Fall (so it was quite cold) we still used it as it was well kept and very clean!
Everything was clean and comfortable, and new towels were provided each day by their housekeeper.
Overall no complaints.
Would definitely stay here again!