[1][2] Barnet was born the son of a builder in Arbroath, Scotland, and was educated at the local high school.

In 1843, at the age of sixteen, Barnet moved to London, where he became a builder's apprentice, studying drawing under William Dyce RA and architecture with CJ Richardson FRIBA.

In that period the department built 169 post and telegraph offices, 130 courthouses, 155 police stations, 110 lock-ups, 20 lighthouses and many other types of buildings.

The Colonial Architect's Office completed this large task in nine months, including preparing 412 drawings, management of the accounts and payments associated with the project, and supervision of the building.

By contrast, also in 1874, the first stage of his General Post Office in Martin Place received high praise, putting aside the much criticised carved figures in the arcade.

[1] Barnet's work drew from a variety classical sources, sometimes with elements from specific buildings, with levels of elaboration or features that suited the function.

For instance his courthouses often included a bold Neoclassical portico, while his post offices often featured a rustic Italianate clocktower, and both types of buildings often featured generous verandahs or arcaded loggias, shaded areas that suited the hot colonial climate.

His larger works like the Sydney departmental buildings, the GPO, and the Garden Palace synthesised elements from various periods of the Italian Renaissance, as well as from English architects such as Christopher Wren and Charles Robert Cockerell, to create impressive compositions.

Barnet was also critical of domestic architectural fashions, cluttered with what he saw as useless ornamentation, and "surmounted with blazing red tiles from France."