How could you stay at forest lodge with reasonable price in Korea.

I think you there are many ways like hotel or condominium or B&B which you cold stay at near the forest in Korea. But there is no other way you could really stay in the forest than NFA condominium, National Forest Administration. I stayed at one of NFA condominium at Hanjeungsan last time which I could quietly stayed really inside the forest without any disturbance.  There arr always several reputed mountain near to the condominium which is inclusive of cooking facilities. You could conveniently come back to the condominium as soon as you finished hiking or walking at the forest since the condominium are usually at foot of the mountains. These condominium is equipped with shower and cooking facilities for your own. You do not need to bring any cooking facilities unless you want special cooking facilities. There are about 100 condominium at the mountain forest running by the three different organization, NFA (National Forest Administration or by local government organization or by nominated private organisation. But their condominium and its facilities are almost same since they are following NFA guide for construction and operation conditions. You could get some basic information of these condominium at NFA site but these condominium are independently running  as per NFA guide line. You may check this linked site for basic general information. http://english.forest.go.kr/newkfsweb/html/EngHtmlPage.do?pg=/esh/recreation/information/UI_KFS_0105_010501.html&mn=ENG_05_01_05_01
 I could introduce typical example of Youmyeongsan Condominium which is about 80 km at East from Seoul Downtown taking about one and half and hour by car. If you use public transportation, it would take more than 2 hour from Dong Seoul Bus Terminal via Cheongpyeong Bus Terminal. Enclosed please find NFA Condominium Fee which would be reasonable and  convenient price level comparing to accommodation fee at other private area. They made a group for house room and camping deck. Camping Deck is a deck for arranging camping tent which user should bring their own tent. You could see some idea of each room or camping deck at attached photo.  Please note this is condominium style which use must bring all necessary food or drink or anything they want. There is no shop inside condominium as well as near by outside.

It is the most important thing there is no English system how to book each condominium through each NFA condominium. Booking through NFA WEBSITE in only Korean, no English service available so far. Therefore you should book these condominium through your possible local channel. You could get some more at  http://www.huyang.go.kr/eng/intro_gyeonggi03.jsp