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
After failing to get his hands on any tickets for Arsenal fans for the FA Cup final, one unlucky supporter appealed for Royal assistance.
The Royal London Hospital and Newham University Hospital have been declared inadequate by the Care Quality Commission.
Homes have been evacuated as the Army works to defuse an unexploded World War II bomb discovered near Wembley Stadium.