That would have to be Qantas flight QF63, from Sydney to Johannesburg, and QF64, flying in the other direction.
The flight path takes the aircraft in a great circle route of 11,044 kilometres, dipping towards Antarctica.
In about mid-flight the aircraft passes over the Kerguelen Islands. At that point Perth is 4232 kilometres away.
The closest airport at which the aircraft could land is Durban, on the east coast of South Africa, 3878 kilometres away.
It is for that reason that the aircraft Qantas uses on this route is a Boeing 747, which has a substantial safety margin since the 747 can remain airborne with just one if its four engines operating.
However, new generation aircraft such as Airbus' A350XWB and Boeing's 787 Dreamliner have an ETOPS rating of 5½ hours or more, the amount of time these aircraft are certified to fly on one engine.
This would allow them to operate safely on the Sydney-Jo'burg route, and another reason why twin-engine aircraft are challenging the dominant position once held by four-engine aircraft.
See also: Which airlines fly with the most empty seats