We stayed in Apartment Madrid for three nights in early October 2013 and couldn't fault it for location. A very easy walk from the Pile gate, no steps except to get up to the apartment, and we were met by Slavica who carried our bags upstairs and down again when we left, so no problems there.
The apartment is in a street that runs parallel to the main street, Stradun, but we heard no noise at all during our stay. The bed would have to be one of the most comfortable I have slept in too.
There are several cafés and restaurants right on the doorstep and we ate locally every night, just stepping outside - lovely! Also during our stay there was a folk concert directly across from the apartment so we had free entertainment one night - very pleasant.
We did have problems with the hot water system during our stay and that took a couple of days to resolve, otherwise I would have given five stars - Slavica did her best to sort it out for us and we were given a very nice bottle of wine and some local goodies which were appreciated.
We would definitely return to these apartments if we returned to Dubrovnik as it was just so easy to walk anywhere we wanted to go.
Thank you Bozo and Slavica.