A nurse and a police officer who had been aiding the evacuation were among those killed in the strikes.
The DSNS said three floors of the four-storey building, as well as the roof were partially destroyed.
It said a fire that broke out after the strikes was later contained.
Several residential buildings in the area were also damaged.
Reacting to the Russian strikes, Ukrainian President Volodymyr Zelensky urged the world to “pay attention to what Russia is targeting.
“They are waging war on hospitals, civilian objects, and people’s lives,” he said in a statement on social media.
Russian President Vladimir Putin launched a full-scale invasion of Ukraine in February 2022.
In its latest operational update on Saturday, the Russian defence ministry says its forces carried out strikes on several Ukrainian brigades in the Sumy region.
The defence ministry did not comment on the reported hospital strikes.
Sumy borders Russia’s Kursk region, where Ukraine launched its surprise incursion in August, capturing at least one town and a number of villages.
The apparent aim of the offensive was to halt Russia’s advance in eastern Ukraine by redeploying troops to Kursk.
However, Russian troops have since claimed to have captured several villages in Ukraine’s Donetsk region, and are now threatening a key town of Vuhledar.
Saturday’s strikes come a day after Zelensky met Donald Trump in New York. The Republican presidential nominee has repeatedly criticised the Ukrainian leader on the US campaign trail, and a meeting between the pair had seemed unlikely until hours before.
On Thursday, the Ukrainian leader met US President Joe Biden and Vice-President Kamala Harris at the White House to discuss his “victory plan”, which he hopes will pressure Russia into agreeing a diplomatic end to the war.
Hours before, Biden had announced a further $7.9bn (£5.9bn) package of military assistance to Ukraine.