Unrelated Umbilical Cord Blood Transplantation for Severe Aplastic Anemia and Hypo-plastic MDS Using CordIn(TM), Umbilical Cord Blood-Derived Ex Vivo Expanded Stem and Progenitor Cells to Expedite Engraftment and Improve Transplant Outcome

* INCLUSION CRITERIA - RECIPIENT: * Diagnosed with severe aplastic anemia with bone marrow cellularity \<30% (excluding lymphocytes) associated with RBC or platelet transfusion dependence and/or neutropenia (absolute neutrophil count \<=1000 cells/ uL or for patients receiving granulocyte transfusions, absolute neutrophil count \<=1000 cells/ uL before beginning granulocyte transfusions). OR History of severe aplastic anemia transformed to MDS that meet the following criteria: a) International Prognostic Scoring System (IPSS) risk category of INT-1 or greater, b) \< 5% myeloblasts and \< 30% of cellularity in the bone marrow on screening morphologic analysis. -Intolerance of or failure to respond to immunosuppressive therapy. This also includes patients who have failed immunosuppressive therapy with ATG and cyclosporine or therapy with cyclosporine combined with eltrombopag in those who are intolerant of or do not have access to treatment with ATG. Identification of either a) at least one alternative donor (i.e. HLA- haploidentical related donor (i.e. \>=5/10 HLA match: HLA-A, B, C, DR, and DQ loci) or \>=9/10 HLA matched unrelated donor) who is available to serve as a stem cell donor for a salvage allogeneic transplant in the event that the CordIn(TM) unit has been rejected or b) umbilical cord blood unit/s that can be used for a salvage cord blood transplant in the event that the CordIn(TM) unit has been rejected. -Availability of at least one \>=4/8 HLA matched (HLA-A, B, C, and DR loci) cord blood unit from the National Marrow Donor Program (NMDP). The cord blood unit must contain a minimum TNC of at least 1.8 x 10\^9 and at least 1.5x10\^7/kg TNC and at least 8 x 10\^6 CD34+ cells (all doses prior to thawing). Exception: Cord units containing at least 8 x 10\^6 CD34+ cells but less than 1.8 x 10\^9 TNC may be eligible for use on this trial if 1. the pre-expansion CD34/kg recipient cell number is at least 1.5 x 10\^5 /kg AND 2. approval for use of this cord unit for expansion is granted by Gamida Cell. 3. the final cell counts on the cultured and non-cultured fractions meet the IND specified minimal release criteria. * The CBU will have undergone volume reduction (both plasma and red blood cell depletion) prior to cryopreservation. All CBUs should be procured from public banks that meet local applicable regulations. * Ages 4-60 years inclusive. * Ability to comprehend the investigational nature of the study and provide informed consent. The procedure will be explained to subjects aged 4-17 years with formal consent being obtained from parents or legal guardian. EXCLUSION CRITERIA - RECIPIENT (ANY OF THE FOLLOWING): * Availability of an HLA identical (12/12) matched related or unrelated donor who is available within optimal timeline and suitable considering graft source and established donor selection factors (e.g. age, sex, viral exposure, ABO compatibility, pregnancy status, etc) per PI discretion. * ECOG performance status of 2 or more. * Major anticipated illness or organ failure incompatible with survival from transplant. * Current pregnancy, or unwillingness to take oral contraceptives or use a barrier method of birth control or practice abstinence to refrain from pregnancy, if of childbearing potential for one year. * Diagnosis of Fanconi s anemia (by chromosome breakage study). * Diffusion capacity of carbon monoxide (DLCO) \<40% using DLCO corrected for Hgb or lung volumes (patients under the age of 10 may be excluded from this criterion if they have difficulty performing the test correctly and thus are unable to have their DLCO assessed). * Left ventricular ejection fraction \< 40% (evaluated by ECHO). * Transaminases \> 5x upper limit of normal. * Serum bilirubin \>4 mg/dl. * Creatinine clearance \< 50 cc/min/BSAm\^2 by 24-hour urine collection adjusted by body surface area. * Serum creatinine \> 2.5 mg/dl * Presence of an active infection not adequately responding to appropriate therapy. * History of a malignant disease liable to relapse or progress within 5 years. * Allergy to bovine, Gentamicin, or to any product which may interfere with the treatment. * Presence of donor-specific antibodies (DSA) to the umbilical cord blood unit and for cohort 1, to the haplo-identical donor.