One degree of longitude at the equator is approximately 111 km. As we move north or south the size of a degree changes. What is the distance that each degree of longitude has at the poles?

ALL of the meridians of longitude converge (come together) at the North and South poles.  So at a pole, the distance between different longitudes is ZERO.

if you're standing at the North or South pole, you're located at EVERY longitude, because any other longitude is zero distance from you.

You're also located at NO longitude.  Because . . .

==> If you're standing at the North pole and you take one step, it doesn't matter what direction you stepped, you've moved South. All directions from the North pole are South.

==> If you're standing at the South pole and you take one step, it doesn't matter what direction you stepped, you've moved North.  All directions from the South pole are North.

an obstruction

none, its a machine