First of all, please let me explain what is Tokyo.
Tokyo is not a city in its traditional meaning, Tokyo is a whole prefecture and at least half of it is covered by evergreen forests and scenic mountains. Quite unexpected for Tokyo, right?
- There are bears in Japan and even in Tokyo. But odds of meeting one are quite low, so I would not worry. Monkeys are much more of nuisance.
- This is fine as most Tokyoites don't have car anyways.
- Then, probably, we have to exclude Yamanashi, Gunma, Nagano - very close to Tokyo, very scenic and quite rural... I would offer Karuizawa which lies in Nagano prefecture, but there are only some lakes. Yamanashi prefecture got famous Fuji lakes, but it is landlocked otherwise.
- Impossible. Japan is very religious country and temples or religious institutions are virtually everywhere.
These are places I would highly recommend, but they fail by at lest one of criteria's you set:
- The most isolated place would be the western side of Izu peninsula. It is very rural, nature is breathtaking, quite wild, but trains don't go there. Car is the only option.
- Yamanashi prefecture. It is Tokyo prefecture neighbour, so it is very close. There are breathtaking views of Fuji mountain, five Fuji lakes, incredibly beautiful flower fields and much-much more. But it is landlocked, no sea is there.
- Nagano and Gunma - depends on the place, but really the wildest mountains you can imagine with places where no human foot stepped in. Trains do go there, and buses can be used for local commuting, but these two prefectures are landlocked too.
- Kanagawa prefecture - Shonan, Kamakura and Miura peninsula are within my Top5 of Japan's most beautiful places. Trains do go there, it is fairly close to Tokyo (in fact still within Bigger Tokyo Area), it is next to the beautiful sea, but you will see temples and pagodas on the every corner, because Kanagawa got very rich history.
- Isehara city in Kanagawa, Ooyama mountain - very close from Tokyo, access is perfectly good - just ride train to Isehara and bus from there to Ooyama entrance. The mountain is huge, nature is scenic, views from there are breathtaking. There are some bears sometimes and huge temple in the middle of the path.
So what do we have left?
- Atami. It is hot spring town in the northern part of Izu peninsula with scenic beaches. Quite mountainous with plenty of nature and wilderness. Trains go there (about an hour from Tokyo), but the town itself is sleepy and frozen in time. Worth to go, but not really the best place to go.
- Boso peninsula. Although geographically very close to Tokyo, access there is difficult - whether you have to ride slow, local trains around Tokyo bay, or go to Miura peninsula and ride ferry from there. It is quite wild, but remember that places accessible by public transport cannot be off the beaten path. I have been there several times and although Boso peninsula is very beautiful it is quite boring too.
- Hakone - great place with sea besides and warm lake in the volcanic crater. Hot springs, mountains and much more. Access is very easy - ride train to Hakone-yumoto, and take bus to Motohakone from there, or mountain cablecar to any station you wish - all of them are interesting.
- Southern islands of Tokyo aka Seven Izu Islands. Incredibly beautiful, still within 300 kilometers, accessible whether by boat from Takeshiba pier in Tokyo (JR Hamamatsucho station) or airplane from Haneda airport (Hachijojima island) or Chofu airport (other islands). There are no habu (yet) despite popular belief, by the way.
Update: according to islanders, Hachijojima have had a habu population, but introduction of weasels solved the problem.