Yes, countries in the Sinosphere have historically used Chinese characters to write their languages. That's why Korean "yaksok" and Japanese "yakusoku" sound so similar. Both words are written with the same Chinese characters, "約束". The characters were borrowed from Chinese, but each language adapted them to its own pronunciation system.
For example, "library" is pronounced "tu-shu-guan" in Chinese, "do-seo-gwan" in Korean, and "to-sho-kan" in Japanese. All three can be written with the same characters, "圖書館". In modern Korea, though, people use Hangul, so very few Koreans actually know how to write "library" in Chinese characters. In Japan, Chinese characters are still heavily used, but for difficult ones, they often write kana alongside them as a reading aid.
It's very much like how Latin "universitas" became "university" in English, "universidad" in Spanish, and "università" in Italian.