Can $400 (American) pay for rent, food, public transit, basic living long(ish)-term in Melbourne or would that mean you're scraping by?

Depends what the exchange rate is at the time.

... and where and how you want to live. Are you looking at renting a flat (apartment), sharing a house or hostel type accommodation?

The exchange rate is currently about AU$1 = US$0.92, so US$400 = AU$435. The AU$ though is currently forecast to each parity with the US$, perhaps as early as Jan-Feb 2010.

You won't be living the high life on that. A room in a shared house (typical student digs) close to the city may cost $130-150 pw, plus your share of utilities. Take-away food is more expensive here and you'd be advised to do a lot of your own cooking. Beyond that we'd need to know more about your preferred lifestyle.

Interesting that's just a bit more than an Australian Postgraduate Scholarship ($20,000 p.a tax free). Students end up with about $385 per week, which suggests to me it's possible but would be a frugal existence.