You'd usually get away with it with Tigers, White-lippeds or Copperheads (these are Australia's three most cold tolerant snakes) but even with them I'd give a few hours of heat per day, even if only 3 hours or so (3-5 hours of supplemental heat per day works really well for these snakes and other cold adapted reptiles - many people go for either too much heat or too low a constant temperature, both of which are pretty bad strategies).
If you're going to do something, do it properly, not half-heartedly, especially when it comes to animals. There aren't any choices I would suggest keeping without controlled temperatures in a typical Melbourne house. Yes, some may manage to survive, but if doing only what you can get away with is your attitude, get a pet cactus or rock or something. You can keep them extremely cheaply, provide everything they need and want for a tiny amount of money and work, but when it comes to reptiles, temperature is king and must be your #1 priority, it's something you should never skimp on.