How does one determine the beginning and ending of the Sabbath in the Arctic?

If God commands His people to obey something, that command is always possible to carry out. Where a person lives is inconsequential.

In extreme latitudes, such as the far North, the days are divided slightly differently. In the summer, the sun nearly touches the horizon every day. The lowest point is when one day ends and the next begins. Likewise, in the winter months, the sun nearly reaches the horizon once every 24 hours. This point separates each day.

This equates to an early summer day spanning from midnight to midnight, which then drops back quickly over the next six months until winter. During that part of the year, an average day spans from noon until noon. In both cases, the separation of days corresponds with the sun nearly touching the horizon.

One must remember that God revealed the Sabbath to those living in the Middle East, where, like most parts of the world, the sunrise and sunset are distinct.