Well, it happens to be true.
What you may have seen is a film that shows a large number of firearms being destroyed in the late 1990s. The film is real - about 600,000 guns were destroyed. But the devil is in the details.
Following the Port Arthur Massacre of 1996, there were some changes to Australian gun laws - in simple terms there was an effort made to standardise them across the country, rather than every state having very different laws. And there were some additional restrictions.
At the same time, the government funded a voluntary buyback - where they would pay fair market value for any firearm handed in, tax free, no questions asked. Because Australia had restrictive laws on selling secondhand firearms - you had to be able to background check somebody in order to sell to them, and most people didn't have the means to do that - quite a lot of people had a lot of surplus firearms sitting around doing nothing. Suddenly they could get ready cash for those easily - so a lot of people did. And a lot of them actually used the money to buy more modern firearms. Most of the weapons bought up and destroyed in the buyback were still fully legal - and that's what is seen in the big piles of weapons.
Even recently a huge story about a family with a huge gun collection way out in nowhere where the Aussie gov. found them, took their collection - even of many harmless ancient non-working guns - and they are facing major jail time.
I don't know the story, but if they were 'way out in nowhere', there's a good chance they hadn't bothered to get the right licence, and the weapons weren't registered - and both of those things will get you into a lot of trouble. But if you are licenced, and you register your weapons, it's a different matter.
These are my two main firearms (or rather photos of the same types):
I own both completely legally. The handgun requires me to have a Category H licence, and the SLR, a Category C licence. This is more than the most basic A/B licence (which basically allows hunting rifles and shotguns) but neither were all that hard to get. You just have to provide a reason you need it (the most common reasons are farming/pest control, but there are others as well).
This is the website for my local gunshop. It's a pretty small shop.
This is the website for a somewhat bigger operation.
There are quite a lot of gunshops around - though not as many as there are in the US - that's probably the clearest indication that guns aren't banned. Shops openly selling them located in every decent sized town, and on the web.
And yes, I'm an Aussie, born and bred.