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
The number of millionaires living in London has shot up in the part five years but other parts of the country are catching up.
A three-year-old girl had to undergo surgery to remove an air rifle pellet after she was shot in a west London park.
There has been a sharp rise in the use of bailiff amongst London's local authorities.