Yes, you can remove it yourself in the state of Victoria, although the other states would rely on State laws. In Vic, as long as you're not getting paid to do the removal (eg: It's your own home, or you're owner building) then you can remove it yourself, so long as you use a licensed disposal bin company. The only time you need a license is if you're providing asbestos removal services, so tradesman or demolition companies would need certification since they are being paid to remove it.
There are two main forms of asbestos, Friable and Non Friable. The Friable is the worst type of Asbestos, and if you crushed it in your hand it would turn to dust and is extremely dangerous and easily airborne. The Non-Friable, which is also known as cement sheeting like on old homes, is usually 90% cement with 10% Asbestos fibre (Roughly speaking). It's still dangerous, but being combined with cement makes it less easily airborne than the Friable stuff. In either case you should know what you're getting yourself into and they type you're dealing with. Use all the right equipment (Gloves, Coveralls, Respirator), wet it down throughly, have plenty of gear to wrap it in safely for removal, and possibly have a chat to the EPA before hand to protect your health and neighbours too.