Your clues are minimal, but because it's a .303, I think the "RE" in the brass disc in the stock means, "Royal Engineers". If it was a Carcano, "RE" would mean, "Regio Ejercito", I think.
You probably have some variant of the Lee-Enfield or the earlier Lee-Metford service rifle, which was used by your country for many decades. With your poor description, I can't tell whether it is in original condition, or was cut down to make a sporting rifle. Because it has an aperture rear sight, it's probably a No. 4 Mk. I or II. It could be the No. 5 "Jungle Carbine", too. Doesn't your library or book dealer have gun books with photos?
The ammunition is still common there, I think. Some years ago, the NZ govt. even issued free .303 ammo to those willing to cull excess stocks of red deer.
Any good gun store there should be able to tell you what it is, or any veteran old enough to have used it before the adoption of the 7.62mm SLR replaced it.
If it's in good condition, thank your late mate. It is a collectible rifle, with a lot of history behind it.
Lone Star
P.S. I ran into a New Zealander in a gun shop in Texas just last week! He said there are still plenty of deer there! And that he doubts that your gun laws will get as bad as Australia's...