At the end of last year Kakao opened the Kakao Friends Concept Museum Seoul in the Hongdae neighborhood. I'll let the pictures speak for themselves as to what to expect. The museum is located in the basement of the building that houses their flagship store in Hongdae. The cost of admission is currently KRW 3,000 and you must make reservations. Each admission ticket also includes a coupon for a KRW 3,000 discount at the gift shop so basically admission is "free".
Visit the link here for tickets and more information: http://kfmuseum.com/en/etc/introduction
Here is a map where the museum is located:
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