Yes, it is possible. But only if you use a third party service to upload the advert for you. Unlike it's major competitor domain.com.au, realestate.com.au (REA) is strictly an accredited Agent access site only. This means that private advertisers are unable to access it directly, and when using a third party service, cannot have their private contact details advertised.
The workaround involves finding a licensed Agent company that specialises in posting advertisements on behalf of private sellers and landlords. There are quite a few to choose from, all offering a roughly similar service, with varying levels of tech to get the enquiries to you.
There is a wide range of prices charged however, so it pays to shop around. Some of the longer established providers charge a hefty fee, and have a lot of listings displayed, but given that the aim is simply to get it on REA's site, a lower priced service will do the same job, provided it is there until sold or let, and with no monthly or other extra fees to pay.
If you are confident to sell your own home privately, then the substantial savings you will achieve by not having to pay an Agent's commission will be well worth it.