My guess is its a home made case made into an 8mm x 06.
Converting 8 x 57mm rifles into 8mm x 06 was popular in the years after World War Twice as 8mm rifle ammo wasn't easy to get.
there is also the possibility it is a 333 OKH, .338/06, or even a 400 Whelen if the bullet diameter is as large as you say.