I agree with Badger Arms. The .376 Steyr is designed to approximate .375 H&H ballistics on a short .308 length action. The factory ammo is expensive and uncommon compared to more well established calibers of similar power and configuration. Putting it on a .30'06 length action negates any advantage in size or weight gained by using the short rifle action.
If you are really set on getting a .376, by all means look into it. I have exotic tastes in weaponry as well, so there is nothing wrong with building up a rifle in that caliber for the sheer reason that you want one. I'd love a 9.3x64 Brenneke (which the .376 Steyr is based off) if I could find a reliable source of reasonably priced brass.
If you are just looking for something want approximating its ballistics, might I suggest a .375/.338 AKA the .375 Taylor? Its basically a .338 Winchester Magnum necked up to .375. Brass can be formed by necking up .338 Win Mags (big surprise there!) or necking down .458 Winchester brass, both of which are cheaper and more available than the .376.
It can propel a 270 grn bullet at 2,800+ fps, and a 300 grn solid to 2600fps. That performance puts it far ahead of the factory loaded .376 ammo, and on par with published .375 H&H specs.
While you would need to have the bolt face opened up to belted magnum head diameter, the cartridge is the same length as the Winchester short magnum series so it will work in the .30'06 length action.