In Gangwon Province you can stay at Baekdamsa, which might be a little closer to Seoul and is newsworthy in it's own right.
Many years ago, an ex-president (Chun Doo-hwan) was charged with corruption and took a sort of coward's way out - he sought asylum at Baekdam Temple until people calmed down. This story has no current significance, by the way.
Read about other temples offering overnight stays in the Korea Herald. Read about my temple stay here. GI Korea also visited the temple.
UPDATING my update: Naksan does offer Temple Stays. I will try to find contact information as I have misplaced the info I was given.