We were a couple travelling with a baby. We chose AthINN because it said they had a travel cot, a lift, balcony and great location.
The location is perfect. It's right in the middle of Athens, close to all the sights, bars and restaurants. The room is clean and there were towels, tea/coffee and basic kitchen equipment. The bed is very comfortable.
Although a travel cot was listed and confirmed when we booked, this was not in the room when we arrived and our messages on Airbnb were not answered. However, when we phoned the host, he arranged for a travel cot to be brought to the room.
There is a lift but you have to go down a small flight of stairs to access it. This wasn't helpful with our pram.
There is a terrace/balcony but it is separate from the room and accessed from the main building hallway.
If you are a couple travelling in Athens, this room is great, however it wasn't ideal for our needs. The amenities listed was misleading.