The Volganeft-139, the tanker, was on its way from the port of Azov in the southern Russian region of Rostov to Kerch in Ukraine's eastern Crimea when high waves broke its hull at around 0445 (0145 GMT) on Sunday, Russian media reported.
The tanker, designed primarily for rivers and in service since 1978, was carrying 4,000 tonnes of fuel oil in total.
Crew members adrift
Thirteen crew members were drifting aboard the ship's stern and several other ships are reportedly in trouble in the Kerch Strait, a waterway running between the Azov and Black Seas.
Efforts to reach them were hampered by the storm, which is said to be worsening.
Russia and Ukraine have set up a joint crisis centre to deal with the situation, and an aircraft and helicopter were on standby to fly to the area as soon as the weather allows.
Oleg Mitvol, deputy head of Rosprirodnadzor, said: "The wind is now blowing in the direction of Ukraine's coast, so it is our common problem.
"This problem may take a few years to solve. Fuel oil is a heavy substance and it is now sinking to the seabed.
"This is a very serious environmental disaster."
The likely effects of the spill were not immediately clear. A spill over 700 tonnes is considered large, but the biggest ones run into the tens or even hundreds of thousands.
Almost at the same time as the Volganeft-139 was destroyed, a freighter carrying 2,000 tonnes of sulphur sank in the same storm, off the port of Kavkaz overlooking the Kerch Strait from the Russian side.
Its crew of nine had been rescued after drifting in a raft for a few hours.
"Sulphur is a very inert chemical, and we hope that in the water it will not form any substances dangerous to humans," Mitvol said.
Several hours later, another freighter carrying sulphur sank off Kavkaz, the regional port administration said.
Three of its crew had been rescued by a Ukrainian ship.
The fate of the other eight was unclear. Meanwhile, a Georgian and Turkish freighter were also stranded off Novorossiisk, Vladimir Yerygin, the port chief said.
Yerygin said both crews were safe.