Severe Aplastic Anemia (SAA) is a rare condition in which the body stops producing enough new blood cells. SAA can be cured with immune suppressive therapy or a bone marrow transplant. Regular treatment for patients with aplastic anemia who have a matched sibling (brother or sister), or family donor is a bone marrow transplant. Patients without a matched family donor normally are treated with immune suppressive therapy (IST). Match unrelated donor (URD) bone marrow transplant (BMT) is used as a secondary treatment in patients who did not get better with IST, had their disease come back, or a new worse disease replaced it (like leukemia). This trial will compare time from randomization to failure of treatment or death from any cause of IST versus URD BMT when used as initial therapy to treat SAA. The trial will also assess whether health-related quality of life and early markers of fertility differ between those randomized to URD BMT or IST, as well as assess the presence of marrow failure-related genes and presence of gene mutations associated with MDS or leukemia and the change in gene signatures after treatment in both study arms. This study treatment does not include any investigational drugs. The medicines and procedures in this study are standard for treatment of SAA.