Fludarabine Phosphate, Melphalan, Total-Body Irradiation, Donor Stem Cell Transplant in Treating Patients With Hematologic Cancer or Bone Marrow Failure Disorders

* First remission lymphoblastic lymphoma, or small, non-cleaved cell lymphoma or mantle cell lymphoma

Hodgkin disease:

* Relapsed or refractory after front-line therapy

Failed prior autotransplant

Age \>= 3 and =\< 75 years for blood and bone marrow transplants and age \>= 3, \< 60 for cord blood transplants

No serious uncontrolled psychiatric illness

No concomitant active malignancy other than non-melanoma skin cancer

Non-pregnant and non-nursing women (women or men with reproductive potential should agree to use an effective means of birth control)

Patients may have received prior autologous bone marrow transplant (BMT) or prior myeloablative allogeneic BMT (at least 60 days have elapsed)

At least 2 weeks since prior chemotherapy, radiation treatment and/or surgery

Informed consent

DONOR: Permissible HLA matching: Related donors- single antigen mismatch at HLA A, B or DRB 1; unrelated donors- a single antigen mismatch at HLA A, B, or C, +/- additional single allele level mismatch at A, B, C or DRB1; cord blood \>= 4 out of 6 antigen match at HLA A, B, DRB1)

DONOR: Compatibility at the four most informative HLA loci: A, B, C and DRB1 are important for reducing the risk of GVHD and successful transplant outcomes; the A, B, C and DRB1 loci comprise 8 possible alleles (a haplotype being inherited from each parent); one additional locus, HLA-DQ, is also typed to ascertain haplotypes and assist in the search for a compatible donor; however mismatching at DQ has not been shown to be associated with adverse outcomes; high resolution molecular typing (at the allele level) is now the standard of care for unrelated donor searches and allows greater refinement of the search strategy

DONOR: Matched related donor: a single antigen mismatch at A, B, or the DR transplant from a family member is associated with a higher risk of GVHD but similar overall survival when compared to full identity at these 3 regions; related donor/recipient pairs must be matched at 5 of 6 HLA antigens (A, B, DRBl)

DONOR: Unrelated donor: when evaluating patients for unrelated donor transplant, a higher degree of matching is preferred due to minimize the risk of GVHD; the A, B, C, DRB1 and DQ loci, comprising 10 possible alleles, will be typed routinely for all unrelated transplants; given the higher risk of TRM in mismatched transplants, RIT is often the best way to mitigate the risk; evolving data from the National Marrow Donor Program now makes it possible to estimate the risks of donor-recipient HLA mismatch at the allele or antigen level; the higher risk from HLA-mismatching must be carefully assessed with respect to the clinical urgency and the patient's risk by the transplant physician; antigen level mismatches at DQ are inconsequential to transplant outcomes and are ignored with respect to donor selection for the purposes of this protocol, with matching requirements confined to the 8 loci involving HLA A, B, C and DRB1; for the purpose of this protocol, a single antigen mismatch at HLA A, B, or C, with or without additional single allele level mismatch may participate in this protocol for voluntary unrelated donors (blood or marrow); patients must be at least antigen-level matched at DRB1

DONOR: If a patient has no suitable family donor matched for 5 of 6 HLA antigens (A, B, DRB1) and no suitable unrelated donor is identified or for reasons of urgency, the patient can be considered a candidate for cord blood transplant, provided a cord blood donor is identified with a \>= 4 out of 6 antigen match at HLA A, B, DRB1 antigens; the cord blood product must provide a minimum of 2 x 10\^7 nucleated cells/kg, test negative for HIV and Hepatitis A, Band C, and sterility assays have no growth; the cord blood products are located through the National Marrow Donor Program, the American Registry, or the Bone Marrow Donor Worldwide or other established registries, and may be stored in the N.Y Placental Cord Blood Bank, the St. Louis Cord Blood Bank, or any of the established, registered International blood and marrow banks

DONOR: Donor must be healthy and have nonreactive test results for all infectious disease assays as required by state and federal regulations; donors who screen seropositive for hepatitis and/or syphilis must be cleared by infectious disease consultation

DONOR: The donor must have no uncontrolled cardiopulmonary, renal, endocrine, hepatic or psychiatric disease to render donation unsafe

DONOR: The donor must be able to give informed consent for peripheral blood stem cell collection or bone marrow collection

DONOR: Donors who have poor peripheral venous access, may require central venous line placement for stem cell apheresis

Uncontrolled central nervous system (CNS) disease (for hematologic malignancies)

Karnofsky (adult) or Lansky (for =\< 16 years) performance status =\< 50%

Diffusing capacity of the lung for carbon monoxide (DLCO) less than 40% predicted, corrected for hemoglobin (Hb) and/or alveolar ventilation

Cardiac: left ventricular ejection fraction less than 40%

Bilirubin \>= 3 x upper limit of normal

Liver alkaline phosphatase \>= 3 x upper limit of normal

Serum glutamic oxaloacetic transaminase (SGOT) or serum glutamic pyruvate transaminase (SGPT) \>= 3 x upper limit of normal

Child's class B and C liver failure

Calculated creatinine clearance \< 40 cc/min by the modified Cockcroft-Gault formula for adults or the Schwartz formula for pediatrics

Patients who have received maximally allowed doses (given in 2 Gy fractions, or equivalent) of previous radiation therapy to various organs as follows:

* Mediastinum 40 Gy

* Heart (any volume) 36 Gy

* Whole lungs 12 Gy

* Small bowel (any volume) 46 Gy

* Kidneys 12 Gy

* Whole liver 20 Gy

* Spinal cord (any volume) 36 Gy

* Whole brain 30 Gy Enrollment of patients who previously receive higher than allowed dose of radiation to a small volume of lungs, liver, and brain will be determine by the discretion of the radiation oncologist on the study

Uncontrolled diabetes mellitus, cardiovascular disease, active serious infection or other condition which, in the opinion of treating physician, would make this protocol unreasonably hazardous for the patient

Human immunodeficiency virus (HIV) positive

Patients who in the opinion of the treating physician are unlikely to comply with the restrictions of allogeneic stem cell transplantation based on formal psychosocial screening

Females of childbearing potential with a positive pregnancy test

What this study is about