Alemtuzumab, Fludarabine Phosphate, and Total-Body Irradiation Followed by a Donor Stem Cell Transplant in Treating Patients With Immunodeficiency or Other Nonmalignant Inherited Disorders

Inclusion Criteria: * Primary immunodeficiency disorder or other nonmalignant inherited disease (except aplastic anemia and Fanconi anemia) treatable by allogeneic HCT * Patients with pre-existing medical conditions or other factors that renders them at high risk for regimen related toxicity or ineligible for a conventional myeloablative HCT * Donors: Related donor who is human leukocyte antigen (HLA) genotypically identical at least at one haplotype and may be genotypically or phenotypically identical for serological typing for HLA-A, B, -C, and at the allele level for -DRB1 and -DQB1; related donors must be a match or a single allele mismatch at HLA-A, B, and C (at highest resolution available at the time of donor selection) and matched at DRB1 and DQB1 by deoxyribonucleic acid (DNA) typing * Donors: Unrelated donors who are prospectively: * Matched for HLA-A, B, C, DRB1 and DQB1 by DNA typing at the highest resolution routinely available at the time of donor selection * Only a single allele disparity will be allowed for HLA-A, B, or C as defined by high resolution typing (no mismatching for DRB1 or DQB1 is allowed) Exclusion Criteria: * Patients with Aplastic anemia and Fanconi anemia * Patients with metabolic storage diseases who have severe central nervous system (CNS) involvement of disease, defined as intelligence quotient (IQ) score \< 70 * Cardiac ejection fraction \< 30% or, if unable to obtain ejection fraction, shortening fraction of \< 26%) on multi-gated acquisition (MUGA) scan or cardiac echo, symptomatic coronary artery disease, other cardiac failure requiring therapy; patients with a history of, or current cardiac disease should be evaluated with appropriate cardiac studies and/or cardiology consult; patients with a shortening fraction \< 26% may be enrolled if approved by a cardiologist * Poorly controlled hypertension despite anti-hypertensive medications * Patients with clinical or laboratory evidence of liver disease will need to be evaluated for the cause of the liver disease, its clinical severity in terms of liver function and the degree of portal hypertension; patients will be excluded if they are found to have: fulminant liver failure, cirrhosis of the liver with evidence of portal hypertension, bridging fibrosis, alcoholic hepatitis, esophageal varices, a history of bleeding esophageal varices, hepatic encephalopathy, uncorrectable hepatic synthetic dysfunction evidenced by prolongation of the prothrombin time, ascites related to portal hypertension, bacterial or fungal liver abscess, biliary obstruction, chronic viral hepatitis with total serum bilirubin \> 3mg/dl, or symptomatic biliary disease (Patients will be allowed on to the protocol with liver problems if gastroenterology approves the patient for HCT) * Patients who are positive for human immunodeficiency virus (HIV) * Females who are pregnant or breast-feeding * Fertile men or women who are unwilling to use contraceptives during HCT and up to 12 months post-treatment * Patients with fungal pneumonia with radiological progression after receipt of amphotericin formulation or mold-active azoles for greater than 1 month will not be eligible for this protocol (either regimen A or B) * Donors: Identical twin * Donors: Pregnancy * Donors: HIV positive * Donors: A positive anti-donor cytotoxic cross match is absolute donor exclusion * Donors: If a patient is homozygous at a particular loci, mismatching at that loci is not allowed due to an isolated graft rejection vector, i.e., patient A\*0101 and the donor is A\*0101, A\*0201; such a mismatch may increase the risk of graft rejection; if patient and donor pairs are both homozygous at a mismatched loci, they are considered a two-HLA antigen mismatch, i.e., the patient is A\*0101 and the donor is A\*0201, and this type of mismatch is not allowed * Donor: Donor \< 6 months old, \> 75 years old