Answer:
Anyplace on earth that's farther north than the Arctic Circle, or farther south than the
Antarctic Circle, has at least one day a year when the sun doesn't rise, and at least
one day a year when the sun doesn't set. The closer you get to the poles, the longer
these annual periods of no sunrise and no sunset get.
Most of Alaska is south of the Arctic Circle, and has a sunrise and sunset every day of
the year. The Arctic Circle crosses Alaska on the line roughly through Shishmaref, Allakaket,
and Fort Yukon. North of that line is where you can have 24 hours without a sunrise
or 24 hours without a sunset.