A fire service spokesman said the whole of the roof of the museum and library was on fire.
Station Manager Peter Critchell, at the scene, said:
"Flames are coming through the roof and there is thick grey smoke billowing up. Firefighters are still working hard at the scene to try and bring the blaze under control. We now have water jets attacking the fire from the four corners of the building. We evacuated 30 people from inside the museum and firefighters will be making efforts to salvage as many of the artefacts as possible."
More top news
Londoners who have received new electronic gadgets for Christmas are being advised to take simple steps to protect them from thieves.
The Boxing Day sales got off to a flying start with shops reporting a surge in business during the first hour of trading.
Bus strikes are planned for Monday 29 December and Monday 5 January.