If you venture outside of Seoul during your time in Korea (as you should), Gyeongju is the place to go. Korea used to be divided into three major kingdoms (Silla, Baekje, and Goguryeo) until the Silla kingdom conquered the other kingdoms and made Gyeongju the capital. Gyeongju remained the capital of this region for approximately 1000 years, leading to the creation of a number of archaeological treasures that are well worth your time.
When visiting Gyeongju, there is no better place to stay than Sarangchae. It is conveniently located a 5 minute walk from a beautiful royal tomb park and also nearby Cheomseongdae, the oldest observatory in Asia, which has impressive stonework (imagine a Coca-Cola shaped stone bottle). Amidst the scenery of ancient Korean kings, try your hand at sleeping on the floor of an ancient traditional Korean house just like Koreans did back in the day. Share the experience with others by listening through the paper thin walls or better yet, by meeting them in the courtyard where you can share stories over a nighttime campfire.
For more information: http://www.kjstay.com/content_eng/iboard.cgi