Alpha/Beta T-cell Depleted Blood-forming Stem Cell Transplant From Related or Unrelated Donors for Blood Diseases in Children and Young Adults

Inclusion Criteria: * No Human leukocyte antigen (HLA) identical sibling available AND * NO HLA matched unrelated donor available OR urgent need of HSCT precludes time necessary to search for suitable HLA matched unrelated donor AND * Haploidentical donor OR closely matched unrelated donor available and willing to undergo mobilization and apheresis * If subject has genetically confirmed inherited bone marrow failure, related donor must be evaluated for this disorder and testing must be negative. * If subject has sickle cell disease, donor must be unaffected or have only sickle cell trait * Patient must be diagnosed with one of the following diseases or disorders: * Hemoglobinopathies * Sickle Cell Disease for patients ≤ 21 years of age for whom hydroxyurea has been trialed for at least six months, and failed * Thalassemia Major for patients ≤ 21 years of age * Acquired Bone Marrow Failure Syndromes * Paroxysmal Nocturnal Hemoglobinuria with bone marrow failure * Myelodysplastic Syndromes (lower risk) * Severe acquired aplastic anemia * Inherited Bone Marrow Failure Syndromes * Fanconi Anemia * Diamond Blackfan Anemia * Dyskeratosis Congenita and related telomere disorders * Congenital Thrombocytopenia Syndromes * Severe Congenital Neutropenia * Shwachman-Diamond Syndrome * Inborn Errors of Immunity (IEI) or Primary Immune Regulatory Disorders (PIRD) * Wiskott-Aldrich syndrome (WAS) * Chronic granulomatous disease (CGD) * Severe combined immunodeficiency (SCID) * Primary hemophagocytic lymphohistiocytosis (HLH) * Age ≤ 25 years (except patients with hemoglobinopathies) * Life Expectancy ≥ 3 months * Karnofsky (patients \> 16 years)/Lansky (patients ≤ 16 years) index ≥ 60 * Organ Function Requirements * Renal Function * Creatinine clearance or radioisotope Glomerular Filtration Rate (GFR) greater than or equal to 60 ml/min/1.73m\^2 * Liver Function * Total bilirubin \< 3 mg/dL * Alanine aminotransferase (ALT)/ Serum glutamic-pyruvic transaminase; synonymous with ALT (SCPT) ≤ 3 x Upper Limit of Normal (ULN) for age * Cardiac Function * Ejection fraction of \> 40% by Multiple gated acquisition scan (MUGA) or echocardiogram * Pulmonary Function * No evidence of dyspnea at rest * No supplemental oxygen requirement * If measured, carbon monoxide diffusion capacity (DLCO) \> 50% * Willing to use effective birth control method if patient is of reproductive potential * Informed consent obtained (patient or legal representative) Exclusion Criteria: * Pregnant * HIV infection * Uncontrolled, serious active infection at screening * Significant serious intercurrent illnesses * Enrollment in any other treatment study that would interfere with the endpoints of this study according to judgement of Principal Investigator (or PI designee).