There is only one upscale hotel in the area, and a couple of inns. However, there are plenty of minbak and a campsite near Woljeongsa. A larger campsite and more minbak are available at the entrance to Sogeumgang. A better option may be to overnight in Gangneung (only an hour's drive away) and make your way to the mountains in the morning.
Also, the resorts and farms mentioned in this guide offer accommodations (as noted).
Note: This information was accurate when it was published, but can change without notice. Please be sure to confirm all rates and details directly with the companies in question before planning your trip.