Cyclophosphamide, Antithymocyte Globulin, and Total-Body Irradiation in Treating Patients With Severe Aplastic Anemia Undergoing Umbilical Cord Blood Transplant

Inclusion Criteria: * Life-threatening marrow failure of nonmalignant etiology meeting two of the three following criteria: granulocytes \< 500/mm\^3; a corrected reticulocyte count \< 1%; platelet count \< 20,000/mm\^3 * Failure to respond to the best available immunosuppressive treatment protocol by 75 days after initiation of therapy * Lack of an HLA-identical family member or closely matched (9 or 10 of 10 HLA-locus match) unrelated marrow donor * DONOR: Unrelated UCB unit matched for at least 4 of 6 loci * DONOR: Related UCB unit matched for at least 3 of 6 lock * Selection of the UCB unit(s) will be based upon matching or mismatching at HLA-A, B antigen level and DRB1 allele level typing; while HLA-C antigen/allele and HLA-DQB1 antigen/allele level typing are not considered in the matching criteria, if available each may be used to optimize unit selection * Multiple UCB units are allowed to provide sufficient cell dose; when multiple units are selected, the following rules apply: a) the UCB unit with the least HLA disparity will be selected first (i.e., selection priority is 6/6 match \> 5/6 match \> 4/6 match), additional UCB units may be selected to increase cell dose; b) UCB units must be matched to each other for at least 4 of 6 loci; c) each unit must contain at least 1.5 x 10\^7 Total Nucleated Cells per kg recipient weight; d) the total cell dose of the combined units must be at least 3.0 x 10\^7 Total Nucleated Cells per kg recipient weight Exclusion Criteria: * Severe disease other than aplastic anemia that would severely limit the probability of survival during the graft procedure; patients who present with active fungal infections must be treated to resolve this problem before beginning the conditioning regimen * HIV seropositive patients * Patients who have developed clonal cytogenetic abnormalities or a myelodysplastic syndrome (these patients will be considered in separate protocols for myelodysplastic syndrome, etc.) * Patients with paroxysmal nocturnal hemoglobinuria or Fanconi anemia * Patients \> 40 years of age * Related or unrelated cord blood units with \< 1.5 x 10\^7 Total Nucleated Cells per kg recipient weight * Related or unrelated cord blood units without full testing and negative results for hepatitis A, B, C, HIV, HTLV-1, CMV viruses