Low-Dose Total-Body Irradiation and Fludarabine Phosphate Followed by Unrelated Donor Stem Cell Transplant in Treating Patients With Fanconi Anemia

Inclusion Criteria: * Any patient with marrow failure and increased chromosome fragility as determined in the diepoxybutane (DEB) or mitomycin C test * Any patient with Fanconi anemia (FA) with marrow failure meeting the following criteria: * Granulocyte count \< 0.2 x 10\^9/L * Platelet count \< 20 x 10\^9/L * Hemoglobin \< 8 g/dl * Corrected reticulocyte count \<1% * Any patient with FA as determined by DEB fragility, who has life-threatening marrow failure involving a single hematopoietic lineage * Any patient with FA and pre-existing cytogenetic abnormality including hematopoietic malignancy (myelodysplastic syndromes \[MDS\] or acute myeloid leukemia \[AML\]) in remission * DONOR: Unrelated Donors who are prospectively: Matched for human lymphocyte antigen (HLA)-DRB1 and DQB1 alleles (must be defined by high resolution typing); only a single allele disparity will be allowed for HLA -A, B, or C as defined by high resolution typing * DONOR: HLA typing will be performed at the highest level of resolution available at the time of transplant Exclusion Criteria: * Evidence for hematopoietic malignancy in relapse * Heart or lung disease that would prevent compliance with conditioning and GvHD regimen or would severely limit the probability of survival * Human immunodeficiency virus (HIV) seropositive patients * Females who are pregnant or breastfeeding, or unwilling to use contraceptive techniques during and for the 12 months following treatment * DONOR: Donors who by DEB te…