Shoka Sonjuku was a private school founded by prominent Edo-era intellectual Yoshida Shoin. It's in the grounds of Shoin Jinja, a shinto shrine dedicated to Shoin.
Many famous figures from Japanese history went to the school, including Ito Hirobumi, first prime minister of Japan, and Takasugi Shinsaku, a samurai heavily involved in the war which led to the defeat of the Tokugawa Shogunate and the Meiji Restoration. It is known as the place from where came a lot of great men, who changed Japan between late Edo period and the Meiji period.
Fifty square meters in area, the school consists of one large room (eight tatami - floor mats also used in Japan as a measure of room size), a ten-tatami prep room which Shoin added, and a dirt floor, called doma in Japanese. There are some pictures of Shoin and of students who studied at the school: you can see Shoin at the center of first line, and he's easy to spot since this picture is very famous!
As you can see, it is a small but very precious building of the late Edo period. This school was registered as a part of “Sites of Japan's Meiji Industrial Revolution: Iron and Steel Shipbuilding and Coal Mining”, in the World Cultural heritage, in 2015. You might see some present-day students studying and taking notes here, since it's a place they learn about at school now.
I felt that the site was bigger than I'd expected, since I had only seen it in books before. But there is the shrine besides, and some other buildings there. It is a little bit far away from Hagi station, so I'd recommend that you ride a bicycle to get there.
If you do, you can also get to Hagi Castle very easily! You can go to other interesting historical sights, including the Kikuya Family residence, and Takasugi Shinsaku's birthplace. Hopefully, you can enjoy your visit to Hagi city!
If you want to get there more directly, it's about a ten-minute walk from Higashi-Hagi station, one stop from Hagi on the San-in main line.