I haven't been there yet, but after a lot of research and correspondence with the building, I booked a room at Cube Serviced Apartments. They're rated #2 on TripAdvisor under Specialty Lodging.

In February, we're staying at "About Henry's", a 3 bedroom, 1 bath house in North Carlton. http://www.abouthenrys.com.au/
Hope it's as good as it looks and sounds!
Sue

We stayed here a few years ago. It was fairly well located.
And a reasonable price too.
Apartments were good.

http://www.albertheights.com.au/

Muck

I stayed at the Mantra Southbank last year, which is a hotel / serviced apartment set up, so you get a dining room / breakfast room downstairs, and our room had a kitchen, lounge room, bathroom and separate bedroom. I expect there are other configurations. It was very convenient for our small family. It is a short walk to the Yarra, and you can cross the footbridge to Flinders St Station and from there get into the city (main shopping precinct) easily. There is also a small supermarket about 500m down the road, and a parking garage if you bring a car. There is also a small shopping centre nearby (not groceries but restaurants / boutiques) on the banks of the Yarra.

http://www.mantra.com.au/victoria/melbourne-and-surrounds/melbourne/accommodation/hotels/mantra-southbank/

Lavandula

You could try Citadines or Punt Hill Apartments. They are both chains and have a few locations in the city. We haven't stayed at any in Melbourne but stayed at Citadines in France and they were fine.

http://www.punthill.com.au/
http://www.citadines.com.au/#/apart-hotel

Kay

If you haven't booked anything yet you could also try these Melbourne short stay apartments, http://apartmentalkira.com.au/
We were there last year at around that time and had a ball - nice and central to everything and not too pricey.
I've also stayed at Punt Hill and they were good too.