TL;DR - Probably not, there's good public transport and other options.
If you're living in central Melbourne or inner city suburbs like St Kilda, Brunswick, North Melbourne, or even a little further out, you won't need a car for most activities. The public transport network is extensive and well serviced (even if some of the locals like to complain about it!) with trains, one of the world's largest tram networks, and buses.
There are also a range of car sharing services like Flexicar, Green Car Share, and GoGet that can provide you access to a car for the times you need it. A lot of councils around Melbourne have been really supportive of these services and so you can find car locations near most densely populated areas. These services could even help you out to decide if you want a car or not, while you get used to the city.