The vector field is defined at all points except where x = y = 0.

This is precisely all points on the z axis.

So when you say :

You are simply saying that F is defined everywhere but along the z axis.Define the vector field on the complement of the z axis by

F(x ,y, z) = (-yi+ xj)/(x^2 + y^2)

So simply compute the curl as you normally would with 0 k component.

In General ifF=fi+ gj

Then curl F is[dg/dx - df/dy]k

Remember Green's Theorem in the plane ?