I had a week's stay in Robert's Place in Elizabeth Bay, and found the apartment perfect for my needs.
Well furnished + very comfortable bed - plenty of hot water - well-equipped kitchen for self-catering - and very clean and peaceful ...
Excellent deli just couple of minutes down the road + cafes and other shops close by + and most importantly, the city bus stops right outside the door and runs every half hour or so to Central Railway station via the city. The city to airport shuttle bus also runs from the Deli - better value than public transport.
Definitely a good spot for a stay in Sydney.