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Enoshima, Kamakura and Zushi are on the Shonan coast and reachable from central Tokyo in 90 minutes. Enoshima has direct express service from Shinjuku. Kamakura has the giant buddha but you can easily avoid both him and the very large temple at the north end of Kamakura. All have sandy beaches and bustling retail areas.
A bit further down the road is Odawara and Atami, both Shinkansen stops. Odawara is a bedroom community for Tokyo, Atami is a beach / spa town and rather less busy than the Shonan coast.
Even furthur south is Izu Kogen and Shimoda. Much easier to get there by car, rent something in Odawara.
Completely the other direction is the eastern Chiba coastline. Seaworld is in Kamogawa and has very nice beaches.
You won't find bears or poisonous snakes in Japan without considerable deliberate effort. Lots of monkeys around that will grab anything shiny or edible, and hawks will dive-bomb your beach picnic. Eat under a tent where they can't see it.
Non-beach relaxation can be found in Hakone, just up the hill from Odawara. And the Fuji 5 lakes district is popular in the summer as it's a lot cooler up there.
Upvote:2
Jogashima (城ヶ島), at the southern tip of the Miura peninsula, is a decent day trip. Rocky beaches, tidepools with marine life, and a nice hike along a bluff where you can see cormorants. There's a small town with eateries serving fresh seafood.
The Keikyu rail/bus network sells a one-day pass (I think weekend only) that will cover the train and bus fare from various points in Tokyo/Yokohama and allow you to get off anywhere along the way.
Another choice is to take the Tokyo Bay Ferry from Kurihama (also on the Keikyu line), to Kanaya in Chiba. And in your case, avoid the main attraction, Nokogiriyama (鋸山), which is a temple complex built on top of the cliff, and instead walk south along the coast as far as you please. The neighboring town of Hota is very quiet and typical of small seaside villages in the area.