If the United States wanted to get rid of Alaska and Alaska was therefore forced to be an independent nation it would need to rely on the United States or some other nation to supply military defense for it. Alaska of course does have a lot of natural resources to sell, yet I am skeptical about its prospects for defending against Chinese, Russian or Canadian invasions relying just on the national guard.
Alaska would need to become a nuclear power and develop a fleet of nuclear armed Kamikaze drones submarines and aircraft to assure mutual destruction to its attackers. That would be costly and in violation of a few treaties that it hasn’t signed yet though other nations have.
Alaskans could stay alive until they died eating fish and buffalo or caribou. Man does not live by bread alone though, and Alaska would need some kind of immigration policy to attract more quality people economically speaking to live there and the harm to the environment would be substantial with the increased population that presently is fewer than 800,000 people.
People can live in very poor circumstances and so could Alaskans although they might fare pretty well for-themselves. When Alaska participates in a trade fair with an advanced nation like China, it has art, carvings of ivory, snowshoes, oil barrels and smoked fish etc. while China has super-computers, super-tankers, jet aircraft and so forth and a million manufactures. Exporting logs used to be a prime Alaska business and then they upgraded to manufacturing log homes to sell to the Japanese and others who like those for some reason.
In a peaceful environment Alaska could get on fine, while in a war environment it just would not have the trillions invested to defeat an enemy that is well founded. Tourists could arrive by the millions annually to spend cash looking at lem sheds in remote rural locations and Hollywood could make brave movies there with harrowing adventures of people fighting and climbing on the last glaciers in north America.