WARNING: CONTAINS DISTRESSING CONTENT
The clip shows the two bears climbing onto the vessel moored off the Vilyuchinsk base, in the far-eastern Kamchatka region.
Gunshots can then be heard, before one of the bears falls into the water.
Many people commented on the footage to condemn what happened.
One person wrote: "Criminals. You have no soul."
Someone else said: "So sad, and so angry about it."
A third added: "You are so cruel!"
Others questioned why the bears couldn't have been scared away or moved on, to avoid having to kill them.
However, a voice behind the camera can be heard in the video saying that the bears would have gone into local communities if they'd only been driven away.
"There's no other way," the voice says.
"If you chase it out, it'll wander into the villages. That's how you fight bears in Kamchatka."
The Russian Navy's Pacific Fleet spokesperson told Interfax that the animals swam across Krasheninnikov Bay and climbed on top of the docked nuclear submarine near the village of Rybachiy.
The spokesperson said a hunting expert and instructor from the military town of Vilyuchinsk had to be called in to kill the bears with a 'specialized hunting weapon'.
Without citing any sources, Telegram news channel Baza reported that the bears were shot because the mother 'was very emaciated and wounded and the cub would allegedly become aggressive without its mother'.
The outlet - which the Moscow Times says is reported to have links to Russia's military and security services also claimed that villagers had seen the bears and had been trying to chase them away for several days.
At least 50 aggressive bears have been killed in 2020 so far, the Kamchatka region's forestry management agency told Interfax.
According to the Wildlife Conservation Society, between 10,000 and 14,000 brown bears live on the remote Kamchatka peninsula, where they sometimes cross paths with humans.
While they usually enjoy feeding on roots, berries and honey, bears can sometimes attack humans if they feel threatened in some way.
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