Fludarabine Phosphate, Cyclophosphamide, and Total-Body Irradiation Followed by Donor Bone Marrow Transplant and Cyclophosphamide, Mycophenolate Mofetil, Tacrolimus, and Sirolimus in Treating Patients With Primary Immunodeficiency Disorders or Noncancerous Inherited Disorders

Inclusion Criteria: * Primary immunodeficiency disorder or other nonmalignant inherited disease (except 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 conventional myeloablative HCT and who do not have HLA-matched related or unrelated donors * Patients with a related donor who is identical for one HLA haplotype * Acquired aplastic anemia: severe aplastic anemia (SAA) is defined as follows: * Bone marrow cellularity \< 25%, or marrow cellularity \< 50% but with \< 30% residual hematopoietic cells * Two out of three of the following (in peripheral blood): neutrophils \< 0.5 x 10\^9/L; platelets \< 20 x 10\^9/L; reticulocytes \< 20 x 10\^9/L * SAA diagnostic criteria may be applied to assessment at initial diagnosis or follow-up assessments * DONOR: Related donors who are identical for one HLA haplotype * DONOR: Bone marrow will be the only allowed stem cell source Exclusion Criteria: * Fanconi anemia * Suitably HLA-matched related or unrelated donors * 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 \< 26%) on multiple-gated acquisition (MUGA) scan or cardiac echocardiogram (echo), symptomatic coronary artery disease, or 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 of \< 26% must be seen by cardiology for approval * 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 \> 3 mg/dl, or symptomatic biliary disease * Positive for human immunodeficiency virus (HIV) * Females who are pregnant (beta-human chorionic gonadotropin positive \[beta-HCG+\]) 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) * DONOR: Donor-recipient pairs in which the HLA-mismatch is only in the host-versus-graft (HVG) direction; patients are homozygous and donor is heterozygous * DONOR: Donors who are not expected to meet the minimum target dose of marrow cells (1 x 10\^8 nucleated cells/kg recipient ideal body weight) * DONOR: HIV-positive donors * DONOR: A positive anti-donor cytotoxic cross match is absolute donor exclusion * DONOR: \< 6 months old and \> 75 years old