The strike on Tuesday occurred in the city of Poltava, the capital of the region of the same name, officials said.
Poltava is about 350km southeast of Kyiv.
The city is on the main highway and rail route between Kyiv and Ukraine's second-largest city, Kharkiv, which is close to the Russian border.
The strike appeared to be one of the deadliest carried out by Russian forces since the war began more than 900 days ago on February 24, 2022.
I received preliminary reports on the Russian strike in Poltava. According to available information, two ballistic missiles hit the area. They targeted an educational institution and a nearby hospital, partially destroying one of the telecommunications institute's buildings.… — Volodymyr Zelenskyy / Володимир Зеленський (@ZelenskyyUa) pic.twitter.com/TNppPr1OwFSeptember 3, 2024
"One of the buildings of the (Poltava Military) Institute of Communications was partially destroyed. People found themselves under the rubble. Many were saved," Zelenskiy said in a video posted on his Telegram channel.
"All necessary services are involved in the rescue operation," he said.
He said he had ordered "a full and prompt investigation" into what happened.
Zelenskiy did not provide further details.
The missiles hit shortly after the air raid alert sounded, when many people were on their way to a bomb shelter, Ukraine's defence ministry said, describing the strike as "barbaric".
Rescue crews and medics saved 25 people, 11 of them dug out from the rubble, a defence ministry statement said.
Poltava Governor Filip Pronin announced three days of mourning starting on Wednesday.
"A great tragedy for Poltava region and entire Ukraine," Pronin wrote on his Telegram page.
"The enemy certainly must answer for all (its) crimes against humanity," he said.
The strike came on the day Russian President Vladimir Putin visited Mongolia.
There was no indication that his hosts would heed demands to arrest him on an international warrant for alleged war crimes.
Zelenskiy repeated his appeal for Ukraine's Western partners to ensure swift delivery of military aid.
He has previously chided the US and European countries for being slow to make good on their pledges of help.
He also wants them to ease restrictions on what Ukraine can target on Russian soil with the weapons they provide.
Some countries fear that hitting Russia could escalate the war.
"Ukraine needs air defence systems and missiles now, not sitting in storage," Zelenskiy wrote in English on Telegram.
"Long-range strikes that can protect us from Russian terror are needed now, not later.
"Every day of delay, unfortunately, means more lost lives."
