The museum commemorates the 1853-1856 Crimea War between the Russian Empire and a coalition that included the Ottoman Empire. Russia was defeated in that war.
Sevastopol's Russian-installed governor, Mikhail Razvozhayev, said on Telegram the museum's roof was hit. He did not provide details about the damage or whether there were any casualties.
"The enemy will pay for this sacrilege!" Razvozhayev said in his post early on Wednesday.
Elsewhere in Crimea, authorities cut train schedules for night hours, the peninsula's Russian-installed governor Sergei Aksyonov said on Telegram, after a drone attack this week injured a train driver and killed his assistant.
The Black Sea peninsula of Crimea, annexed by Russia from Ukraine in 2014, is facing a fuel shortage following recent Ukraine drone attacks just as the holiday season starts.
Ukrainian President Volodymyr Zelenskiy last week proposed face-to-face talks with Russian President Vladimir Putin which he rejected. After the train incident, the Kremlin said Ukraine was undermining efforts to find a peaceful resolution to the conflict.
Elsewhere, the city of Novokuibyshevsk in Russia's Samara region, a major oil hub on the Volga river that hosts several refineries operated by the state-controlled oil giant Rosneft , was repelling drone attacks, regional governor said.
Authorities urged residents of the city of one million people to seek shelter as public transportation was suspended amid air raid alerts, local media reported.
Continuing attacks by Ukraine on Russian energy infrastructure has forced Moscow to cut its oil output, the world's third-largest.
In the southern Rostov region of Russia bordering Ukraine, falling debris from a drone triggered a fire in a fuel tank at a civilian site, the regional governor wrote on Telegram.
Also on Telegram, the mayor of Moscow said the city was repelling drone attacks.
In rare moves, remote Russian oil-producing regions Khanty-Mansiysk, Perm and Tyumen, as well as industrial regions Chelyabinsk and Sverdlovsk in the Ural mountains thousands of kilometres from Ukraine, issued air raid alerts, according to social media posts by local authorities.
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