This "war" was not an actual war, rather it was a campaign to lower the population of Emus that infested rural farms and communities. People who lived in the Outback found their aggressive nature particularly annoying.
While Australiaball was suffering from the Great Depression, there were tons of emus that were eating his crops and generally pissing him off. Soon Australiaball "declared war" on the Emus. Despite Australiaball's use of machine guns, the Emus proved to be a formidable opponent. Soon Australiaball had to surrender to not be overrun.
The Emus won the Great Emu War, and Australiaball never speaks of it to this day.
Australia is also planning an attack on Cane Toads that were imported by those frickin' [Censored] in the US. The Emus to this day have been supplying Cane toads with Anti-Australian misdirection tactics making them harder for Australiaball to catch.