Would it be Code legal to have an all-240V house in the United States?
Such a fun question deserves an equally fun answer.
Sure you can!  And there's a neat trick that will help with all those troublesome NEC 2014/17 rules: simply make sure to build this house in a country where the NEC does not apply!
Most of the world uses 230V as their standard supply voltage, which is pretty close, but if that's not quite enough for you, there are 19 countries that use exactly 240V by default, and would make an ideal location to build a 240V-only house.

Brunei
Cook Islands
Cyprus
Falkland Islands
Fiji
Gibraltar
Isle of Man
Kenya
Kiribati
Kuwait
Libya
Nauru
Oman
Papua New Guinea
Qatar
St Lucia
Seychelles
Tonga
Uganda

(source: https://en.wikipedia.org/wiki/Mains_electricity_by_country)