I stayed at Vasso's place for 3 weeks and it was awesome. Right in the middle of town but the double glazed windows kept the the noise out perfectly. I could be in the thick of it when I wanted and out of it when I needed.
From here I walked to pretty much every site of interest in Athens but if you'd rather not walk the trains is just 3-minutes walk away.
Next time I'm in Athens I hope Vasso's place is available because I would definitely want to stay there again.