Riding enduro bikes offroad, illegal?

Is it legal or illegal to ride enduro bikes (motorcycles) offroad in Japan? Not on actual tracks but out in the woods?

The "woods" as you say may be a national park or someone's property. I am not certain, but it sounds unlikely.....

The mountains are criss crossed with trails which are legal to ride on with a registered enduro bike, unless marked closed. Just avoid hiking trails and private land, and you will be alright.

I have been engaged in enduro races management long years.

All the land in Japan has owners.
It is basically illegal to enter the land without obtaining permission from the owner.
However, no one will report to the police in an unmonitored abandoned place.
There is a possibility that even in the mountain where no one can see it is reported due to noise.

So, you should ask local motorcycle shops, riders or gas startions directly.
They may know something but there may not be many riders familiar with Off-Road scenes.
http://www.gaijinriders.com/forumdisplay.php?29-Gaijinriders-Forums

There was a problem with the (coin) car wash,
https://youtu.be/9dEllX_Xrhw
and the rider who is interested in off-road bike has become less than 1/10 of the heyday.

(notes)
It is illegal to run public roads with motorcycles
without "license plate(Japanese are calling "number plate")" and more.

And refueling to a portable tank(can) was impossible at "self-(gas)station" by the fire law became strict.
https://www.city.takaoka.toyama.jp/yobo/bosai/shobo/yobo/gasorinkeikou...

There are dedicated road bikes in Japan, which are free to use.
Same as for mountain bike road paths which I use

As mentioned before local bike shop may know, but count on that you need to travel since the dedicated paths are not every where. I often go by car to a dedicated location which takes about 40 min by highway from my house.

@Marshal rider
And refueling to a portable tank(can) was impossible at "self-(gas)station" by the fire law became strict.
https://www.city.takaoka.toyama.jp/yobo/bosai/shobo/yobo/gasorinkeikou...

Maybe you should check once more the website.
It mention that you cannot refile the gasoline can by your self but have to refile it by the employee.
Also you need to have a dedicated gasoline can and not use a plastic jerrycan.

So refilling a gasoline can is possible.