The wooded South of Tokyo
South of Tokyo, in Kanagawa prefecture, lies a quiet wood called Kiyokawa-mura. The rivers run through the steep and forested hills of this district, and it is very popular for both camping and barbecues. There are many options for simply camping, but another popular option is renting out a cottage or bungalow near the campground.
A very fun and easy location is the Kiyokawa-mura "Richland" resort that is a fully established campground with facilities, and even a small restaurant. You can spend the day exploring the nearby woods, and gather around the barbecue areas at night to meet other campers, and even buy roasted fish from the staff. I highly recommend this area for a genuine outdoor lodging experience.