The gang is believed to have entered the property between 01:00 and 02:00 BST.
The stolen items were from the collection of the former British Empire & Commonwealth Museum, which was donated to the Bristol Museum after its liquidation in 2013.
Police said the stolen artefacts had “significant cultural value”.
Following the incident, Bristol City Council released images of an elephant carved in ivory, an ivory statue of the Buddha, a ship lantern and an Emancipation token.
A police spokeswoman said: “We continue to appeal for the public’s help to identify the four men pictured and have released two short video clips of the people we wish to speak to.”
Anybody with information is asked to contact the force.