The FAQ page takes a little searching, but this is what it has.
Q. Does Melbourne Airport have a post office?
A. There is no Post Office in the Terminals however there are two Australia Post Mail Boxes located at Melbourne Airport – one on the Departures Drive (T1 Qantas Departures end) and one on the ground floor of the new T4 Multi Level Car Park.
Unofficially, I’m going to guess that the T1 Australia Post Mail Box is outside of the terminal on the first floor (note: not ground floor). The maps don’t really specify where the T4 Car Park is, let alone where the drop box is.
Purchasing Stamps (and Envelopes)
Envelopes and stamps are normally available at the newsagents located throughout the Terminals.
So this is a thing – Australia Post has postal lockers at the Terminal T4 Car Park. These are used primarily for parcel pickup and drop-off. No word on if they can be used for letters; perhaps this is the T4 Car Park mail box. (If anyone can confirm please let me know!)
Last Updated 10 June 2017