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) = (-y i + x j)/(x^2 + y^2)
So simply compute the curl as you normally would with 0 k component.
In General if F =f i + gj
Then curl F is[dg/dx - df/dy] k
Remember Green's Theorem in the plane ?