See the fuller question for far more. But for a quick answer on Barrow specifically, here's the Star-K:
WINTER:
For example, in Barrow located at the northern tip of Alaska, on December 1 the sun does not rise. However, at 1:15 p.m. there is the most sunlight of the day 30 (theoretical chatzos hayom [=noon]). Therefore, one may daven
Shachris and perform daytime mitzvos between 10:40 a.m. and 1:15 p.m.31 (during these pre-dawn light conditions) and daven Mincha at 1:45 p.m. (1/2 hour after chatzos during the post sunset conditions). Shabbos ends at 4:23 p.m. when the stars come out.
I don't see them addressing when Shabbos starts, specifically. Possibly best to avoid working as early as 1:16PM on Friday, I'd assume. In which case it would be best not to light shabbos candles; by the time it's sufficiently "Friday afternoon" to light shabbos candles, it may already be sunset and prohibited to do so.) So probably light candles, without saying a bracha, before 1:16PM. I don't think their article addressed this one ... anyone?
SUMMER:
This one's a bit easier. When there's no good sunset, we treat sunset as 6PM standard time. Hence, on standard time, the ideal time for lighting candles is 5:42PM, before 6:00PM standard time (likely 6:42PM--7:00PM with daylight saving), but if you wanted you could light candles as early as 4:45PM standard time.
But ...
bring along an almanac from home; if you came from somewhere where sunset in the winter is before 6PM (would have to be South of the equator if this happens in the summer, I think), then start shabbos at that earlier sunset time of your hometown.
shabbattag. MordechaiBenDaniel, note that the other question asks "How is this handled", seemingly asking for, inter alia, actual practice. But very well, I won't close it. If I had a vote, I'd vote to close it, though. – msh210 Jun 15 '11 at 16:19