Fludarabine Phosphate, Cyclophosphamide, and Total-Body Irradiation Followed by Donor Bone Marrow Transplant, Mycophenolate Mofetil, and Cyclosporine in Treating Patients With Fanconi Anemia

Inclusion Criteria: * Any patient with FA and bone marrow (BM) failure involving 2 of the following 3 lineages: granulocyte count \< 0.5 x 10\^9/L, platelet count \< 20 x 10\^9/L, or hemoglobin \< 8 g/dL * Any patient with FA who requires red blood cell or platelet transfusions because of marrow failure * Any patient with FA who has a life-threatening BM failure involving a single hematopoietic lineage * Any patient with FA and pre-existing cytogenetic abnormality including hematopoietic malignancy (acute myeloid leukemia \[AML\] or myelodysplastic syndrome \[MDS\]) in morphological remission (defined as absence of circulating blasts and bone marrow blasts \< 5% as assessed by morphology); Note that hematopoietic recovery is not required for remission status * Patients must have a negative cytotoxic cross match with donor * DONOR: Related, human leukocyte antigen (HLA)-haploidentical donors must be identical for one HLA haplotype and mismatched for any number of HLA-A, -B, -C, DRB1 or DQB1 loci of the unshared haplotype * DONOR: Unrelated, HLA-matched donors must be matched at HLA-A, B, C, DRB1 and DQB1 by deoxyribonucleic acid (DNA) typing at the highest resolution routinely available at the time of donor selection; a single allele mismatch at HLA-A, B, or C is allowed OR a single DQB1 mismatch is allowed * DONOR: Bone marrow will be the only allowed hematopoietic stem cell source * DONOR: Haploidentical donor selection will be based on standard institutional criteria, otherwise no specific prioritization will be made amongst the suitable available donors Exclusion Criteria: * Patients having available HLA-matched related donors * Significant organ dysfunction that would prevent compliance with conditioning, graft-versus-host disease (GVHD) prophylaxis, or would severely limit the probability of survival, such as liver disease/failure (active hepatitis, moderate to severe portal fibrosis/cirrhosis confirmed by biopsy or uncorrectable hepatic synthetic dysfunction), lung disease, or cardiac disease (ejection fraction \< 35%, or if unable to obtain ejection fraction, shortening fraction of \< 26%; if shortening is \< 26% a cardiology consult is required with principal investigator \[PI\] having final approval of eligibility) * Human immunodeficiency virus (HIV) seropositive patients * Fertile females who are unwilling to use contraceptive techniques during and for the twelve months following treatment, as well as females who are pregnant or actively breast feeding * Fertile males who are unwilling to use contraceptive techniques during and for the twelve months following treatment * AML/MDS in morphological relapse, defined as having circulating blasts or bone marrow blasts \>= 5% as assessed by morphology * Active infectious disease concerns * Karnofsky performance score \< 50 or Lansky performance score \< 40 * DONOR: Donors found to have Fanconi anemia based on chromosomal breakage analysis * DONOR: Donors who are not expected to meet the minimum target dose of marrow cells (1 x 10\^8 nucleated cells/kg recipient ideal body weight \[IBW\]) or who are unwilling to be bone marrow donors * DONOR: HIV-positive donors * DONOR: Donors who are cross-match positive with recipient * DONOR: Recipient homozygous at mismatched locus; if the recipient is homozygous at HLA-A, B, or C and the donor is mismatched at that locus, the donor should be avoided; exceptions must be discussed with the PI