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

Background: Severe aplastic anemia (SAA) and myelodysplastic syndrome (MDS) are bone marrow diseases. People with these diseases usually need a bone marrow transplant. Researchers are testing ways to make stem cell transplant safer and more effective. Objective: To test if treating people with SAA or MDS with a co-infusion of blood stem cells from a family member and cord blood stem cells from an unrelated donor is safe and effective. Eligibility: Recipients ages 4-60 with SAA or MDS Donors ages 4-75 Design: Recipients will be screened with: * Blood, lung, and heart tests * Bone marrow biopsy * CT scan Recipients will have an IV line placed into a vein in the neck. Starting 11 days before the transplant they will have several chemotherapy infusions and 1 30-minute radiation dose. Recipients will get the donor cells through the IV line. They will stay in the hospital 3-4 weeks. After discharge, they will have visits: * First 3-4 months: 1-2 times weekly * Then every 6 months for 5 years Donors will be screened with: * Physical exam * Medical history * Blood tests Donors veins will be checked for suitability for stem cell collection. They may need an IV line to be placed in a thigh vein. Donors will get Filgrastim or biosimilar (G-CSF) injections daily for 5-7 days. On the last day, they will have apheresis: Blood drawn from one arm or leg runs through a machine and into the other arm or leg. This may be repeated 2 days or 2-4 weeks later.