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In the Heart of the Sea is primarily based on a famous historical ship, which also was part of the inspiration for Herman Melville's Moby Dick. That ship was named Essex. Launched at Nantucket in 1799, it was lost at sea in 1820 along with most of the crew in a remote part of the Pacific Ocean. It was apparently attacked and destroyed by an angry sperm whale.
As mentioned in a comment by @AaronBrick, the blubber would have been stripped from the whale still in the water next to the ship, and then hoisted one strip at a time onto the deck. The process of removing the blubber from the whale is known as flensing. Blubber was indeed boiled down to liquid oil and stored in large barels on the holds of ships. These ovens for boiling blubber are known as tryworks.
Here is a nice (if simplistic) diagram of how the Essex would have been laid out. As you can see the tryworks was relatively compact but a large section of the hold would have been dedicated to barrels. The barrels would have held drinking water on the way out to sea and whale oil on the way back home. According to that site, the number of barrels was more like 1,200.
If you're interested in learning more about the Essex disaster and its historical context, I cannot recommend strongly enough the excellent documentary Into the Deep from PBS' American Experience series. There are also many good books about early American whaling, including an illustrated edition of the memoir written by a survivor of the Essex disaster, Owen Chase.