One of my favorite occasions (not in this era, and not quite a naval battle, more like its aftermath):
In 1905, after the battle of Tsushima, the surviving russian ships that were fast enough to flee from the japanese fleet after the surrender of the remaining russian ships, made their way to the original objective, Vladivostok, or fled to chinese ports to be interned until the end of the war. The destroyer Bodriy, on board with the survivors of another russian ship, decided to sail into Shanghai, but was hit by a storm, and in the strong upwind, it could barely move forward at full steam. By the time the storm ended, the ship ran out of coal, and the crew set up the emergency sails to try to reach the coast, but the food and water rations were also running low. In the end, the starving crew even drank the water supply of the boilers. A few days later the ship was found by the british cargo steamer Qualin, which towed the destroyer to Shanghai, where it was interned.
Source (hungarian but a lot of good images): http://acelmonstrum.host22.com/csuz.html