The blast was caused by a Russian S-300 rocket, Kharkiv governor Oleh Syniehubov said in a post on the Telegram messaging app.
All of the victims were employees of private Ukrainian postal and courier service Nova Poshta.
In a statement, the company said the air raid siren had sounded just moments before the attack, leaving those inside the depot with no time to reach shelter. It announced that Sunday would be a day of mourning for the firm.
Ukrainian president Volodymyr Zelensky posted a video showing the building with destroyed windows and construction materials strewn across the floor. He described the strike as an attack on an “ordinary civilian object”.
“We need to respond to Russian terror every day with results on the front line. And, even more so, we need to strengthen global unity in order to fight against this terror,” he wrote on social media.
“Russia will not be able to achieve anything through terror and murder. The end result for all terrorists is the same: the need to face responsibility for what they have done.”
Elsewhere in the Kharkiv region, three people were injured in Russian shelling on the city of Kupiansk, Mr Syniehubov said.
The Ukrainian-held frontline city has been at the heart of fierce fighting as both Moscow and Kyiv push for battlefield breakthroughs amid the looming onset of wintry conditions.
Officials in southern Ukraine said on Sunday that the Russian military had used a record number of aerial bombs over the country’s Kherson region in the past 24 hours.
Natalia Humeniuk, a spokesperson for the Ukrainian military’s Operational Command South, said 36 missiles had been recorded over the area, with some villages being hit by several strikes.
It comes as further south in the east, Ukraine has been trying to stop a new push by Russian forces to gain more territory there, amid Kyiv’s gruelling counteroffensive that has continued for months.
Moscow’s drive to capture the town of Avdiivka encountered fierce resistance on Saturday, Ukraine’s military said, with defences bolstered by fortifications erected nearly a decade ago.
“The enemy is becoming more active, but is incurring heavy losses,” General Oleksandr Tarnavskyi, commander of Ukraine’s troops in the south, said on Telegram.
Russia’s defence ministry, in its evening report, made no mention of Avdiivka, but reported strikes on areas outside Bakhmut, a town seized by Moscow’s forces in May after months of battles. Both towns are in the eastern Donetsk region.
Avdiivka has withstood enemy attacks for months. Video footage shows buildings in ruins and streets barely distinguishable.
The town was briefly captured in 2014 by Russian-backed separatists who seized large swathes of eastern Ukraine, but was retaken by Ukrainian forces who built solid fortifications.
The Institute for the Study of War, a US think-tank, said Russian troops had “marginally advanced” near Avdiivka.
Additional reporting by agencies