This phase II trial studies fludarabine phosphate and total-body irradiation with or without alemtuzumab followed by donor stem cell transplant to see how well it works in treating patients with immunodeficiency or other nonmalignant inherited disorders. Giving chemotherapy, such as fludarabine phosphate, a monoclonal antibody such as alemtuzumab, and radiation therapy before a donor stem cell transplant helps stop the growth of abnormal cells. It may also stop the patient's immune system from rejecting the donor's stem cells. The donated stem cells may replace the patient's immune cells and help destroy any remaining abnormal cells.