If any of my friends or family mention wanting to go to Greece, my first words will be: I know the best place to stay!
Rita's place is the best place to stay in Athens. It is in a very safe area and it is extremely close to the Acropolis (there is also an ice cream place nearly downstairs). The views of the Acropolis from the living room and the roof terrace are postcard worthy. We spent every evening on the roof!!
The beds are comfortable and the AC units beat the Athens heat. The kitchen itself is small but well equipped. And there is a large dining table in the living room although we didn't use it because we had our dinners on the roof.
The best part of staying here was Rita. She was very helpful and provided all sorts of useful recommendations concerning transport, food, sights or whatever we asked. She was even welcoming when we had to wake her up at 2am to be let into our apartment!
PS: If you can, use her taxi driver because he is very nice and will give you a little intro to the sights on your way in. Plus the wifi in the car is a great perk.