Density-functional calculations of the electronic structure of (n,n/2) chiral carbon nanotubes filled with a linear chain of Fe atoms were conducted for the first time. It was found that upon encapsulating a chain of Fe atoms, the initially semiconducting nanotubes Fe5@(4,2), Fe5@(6,3) and Fe5@(8,4) became metallic. In the case of the Fe5@(8,4) structure, only the iron chain was conductive. All the considered nanotubes maintained large energy of the magnetic anisotropy, which is characteristic for a free linear chain of iron atoms. The magnetic moment on an iron atom was found to vary nonmonotonically with increasing the chiral index n; for the Fe5@(6,3) nanotube, the magnetic moment reached that of a free linear chain of iron atoms.