HandyBilly: What he said. ----^
There are a few newer marinas right in the heart of Melbourne that I haven't checked, but a thorough survey done 12 months ago says "No" to permanent live-aboards in Port Phillip, except for the one in Geelong. There are a few places in Westernport but even those are being crunched by the local councils.
The city marinas that allow it are geared up for superyachts in the 90+ ft class, which is where the money is. I'm from Melbourne and I find the situation ridiculous, but that's the way it is.
If I were to sail in there, I'd probably anchor off Mordialloc pier and use the dinghy to get ashore. There's a nice little boating community based in the creek and unlike most parts of the bay it isn't totally focused on day sailing and racing. Lots of old timber fishing boats and the boat builder there (Pompei) has basic stuff like paint and bulk rope. Pompei Boat builders Mordialloc
The pier is public and mooring boats alongside is allowed, so you could tie up a dinghy halfway along near the shelter shed and go shopping at the local supermarket or strip shopping centre, which is within 1km. I've seen a 100' motor boat pick up a crowd of people from there to go day cruising, so it's deep enough to bring the yacht alongside if you need to load anything large or heavy.
In warmer weather there are kids diving off the pier (illegally, I might add) but your stuff should be safe during daylight hours as long as you don't take any silly risks like parking it on the beach without a lock attached. There is a haulout facility available in the creek but I'd only use it in a dire emergency; access is difficult and they always seem to be busy.
Fishing is also pretty good with snapper, bay trout (Australian salmon), mullet, garfish, banjo sharks, gummy shark - and the occasional bream and trevally - being caught. You'll need a state fishing licence, which is available on the internet or from the bait shop at the start of the pier. I think it costs about $25/year. An old Greek bloke at the end of the pier called Harry should be able to tell you what to use for bait. :-)
Public toilets (daytime hours) and free electric BBQs are on the shoreline. Cold water showers also available somewhere for bathers, I've never used them so can't tell you where they are.