Dougo Onsen (道後温泉), located in the island of Shikoku, is considered to be one of Japan's oldest onsen, and it is also one of Japan's most beloved and most famous, thanks to its appearance in numerous works of literature, such as Natsume Souseki's Botchan and more recently, as the inspiration for Miyazaki Hayao's Spirited Away. Even on a bright sunny day there's a sense of atmosphere here -- the bath, with its brilliant blue tile mural and large water cistern, commands respect, as do some of the wizened grandfathers who slowly ease themselves into the steaming water. The dressing room and lounging room are worn but not shabby, and on the third floor there's a room devoted to Natsume Souseki, who stayed here. Though the information is primarily in Japanese, it's still well worth a peek.
Getting here requires a long train ride, but those with JR Passes can make use of the wonderful overnight sleeper from Tokyo: the Sunrise Seto (サンライズ瀬戸). You can either sleep on a partitioned, carpeted floor section for free between Tokyo and Takamatsu (in Shikoku), or upgrade for about 10,000 yen to an actual room. Either way, this makes for a fun overnight and leaves you relatively well-rested when you reach Shikoku. Be warned: from Takamatsu, the train to Matsuyama takes another three hours. Once there, however, the bath won't disappoint, and there are plenty of other things to do and see in Matsuyama once you're done with your soak. It's worth noting that you should head to the Dougo Onsen Honkan, not the newer Dougo Onsen resort area. From the main station you can take a streetcar that arrives at Dougo Onsen in about 15 minutes. Enjoy!