The site is used to launch rockets to the ISS
Three out of five segments of the 70-metre high roof of the block, used for assembling and testing space vehicles including the Buran shuttle, gave way at around 0720 GMT.
A specialist team of 30 rescue workers is being flown from Moscow to help recover victims and they are expected to arrive in Baikonur shortly.
Fear of collapse
However, Kairzhan Turezhanov, a spokesman for the Kazakh Emergency Situations Committee, said it was unlikely any would have survived the fall.
"We don't know whether they're alive or not," said a Russian duty officer at Baikonur.
The hangar has been cordoned off because of fears that the walls could collapse and local Kazakh rescuers were prevented from entering because of its instability, Mr Turezhanov said.
The BBC's Nikolay Gorshkov says Moscow has been short of funds to maintain the complex, and parts of it have fallen into disrepair.
The Russian Aviation and Space Agency has set up a special centre to investigate the causes of the incident.
World's oldest
The Baikonur cosmodrome, Russia's main commercial launch site, was built in the 1950s and is the oldest working launch site in the world.
The first manmade satellite to orbit the Earth was launched from here.
The Buran project was initiated in 1976 in response to the US space shuttle programme but abandoned after the collapse of the USSR. It completed one unmanned flight in 1988.
More recently, Baikonur has been used to launch commercial satellites and rockets bound for the International Space Station.
Kazakhstan took possession of the cosmodrome after the disintegration of the USSR. But it lacked the funds and expertise to maintain the site, which was looted by local people who sold sophisticated equipment for scrap metal, our correspondent says.
Russia has been leasing Baikonur from Kazakhstan since 1993, though the two former Soviet countries have on several occasions been at odds over rent payments and accidents during launches.