Inclusion Criteria:

* Voluntary written informed consent
* Patients who are considered for allogeneic transplantation based on their disease risk (see below) but lack matched sibling or unrelated donors or who are unable to proceed to allogeneic transplant within 8 weeks, will be considered for UCB transplantation on this study. Only patients for whom RIC will be considered are eligible. RIC is considered in those older than 45 or younger than 45 with Hematopoietic Cell Transplant (HCT) Comorbidity Index of 3 or higher (HCT) Comorbidity Index can be calculated using the following link: http://www.hctci.org/Home/Calculator
* Patients with acute myeloid leukemia (AML) in CR1 that is not considered favorable-risk (favorable risk is defined as patients with t(15;17)(q22;q21), t(8;21)(q22;q22), inv(16)(p13q22)/t(16;16)(p13;q22), NPM1 mutation without FLT3-ITD, and double-mutated CEBPA58,59), AML in CR2 or subsequent CR, high-risk acute lymphoblastic leukemia (ALL) in CR1, or ALL in CR2 or higher, biphenotypic leukemia defined as coexpression of B-lymphoid and myeloid markers or T-lymphoid and myeloid markers in the blast population60or undifferentiated leukemia in ≥CR1. Myelodysplatic syndrome (MDS)/myeloproliferative neoplasm (MPN) patients with less than 10% bone marrow blasts and no peripheral blood blasts on pre-transplant bone marrow aspirate/biopsy are considered for FluCyTTBI regimen. Chemotherapy sensitive (achievement of at least a partial response according to Lugano classification61) Hodgkin's disease (HD) that relapsed following high-dose therapy. Chemotherapy sensitive (achievement of at least a partial response according to Lugano classification) non-Hodgkin's lymphoma (NHL) patients who relapsed post-high-dose therapy and autologous transplantation. Subjects should be enrolled within 30 days of transplant.
* For ALL, high-risk features are defined using modified Hoelzer risk criteria62, these criteria are:

1. High white blood cell count at diagnosis (ie, \>30,000/microL in B-ALL or \>100,000/microL in T-ALL).
2. Clonal cytogenetic abnormalities - t(4;11), t(1;19), t(9;22), or BCR-ABL gene positivity.
3. Progenitor-B cell immunophenotype (eg, blasts expressing membrane CD19, CD79a, and cytoplasmic CD22).
4. Length of time from start of induction therapy to attainment of CR greater than four weeks.
5. Older age - \>60 years old is high risk, 30 to 59 years old is intermediate risk.
6. MRD - a post-remission bone marrow MRD level ≥10-3 by molecular tests.
* Subjects must be ≥ 18 years old and ≤ 70 years old
* Karnofsky performance status (KPS) of ≥ 70% (Appendix A).
* Adequate hepatic, renal, cardiac and pulmonary function to be eligible for transplant. Minimum criteria include:

* ALT, AST: \< 4x IULN
* Total bilirubin: ≤ 2.0 mg/dL
* Creatinine: ≤ 1.5 x ULN
* EF measured by 2D-ECHO or MUGA scan of ≥ 45%
* FEV1, FVC and DLCO ≥ 50% of predicted value (corrected to serum hemoglobin).
* EKG with no clinically significant arrhythmia.
* Patients should have New York Heart Association (NYHA) Functional Classification, class -1 (ordinary physical activity does not cause undue fatigue, palpitation, dyspnea, or angina pain) or class II (ordinary physical activity results in fatigue, palpitation, dyspnea, or angina pain).
* Patients should be evaluated for fitness for HBO by a hyperbaric oxygen trained medical professional who is not part of the study team prior to starting preparative regimen.
* Women of child-bearing potential should have a negative urine pregnancy test within 4 weeks of starting preparative regimen.
* Women of child-bearing potential and men with partners of child-bearing potential must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 30 days following completion of therapy. Should a woman or partner become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician and the investigator immediately.
* A woman of child-bearing potential is any female (regardless of sexual orientation, having undergone a tubal ligation, or remaining celibate by choice) who meets the following criteria:

1. Has not undergone a hysterectomy or bilateral oophorectomy; or
2. Has not been naturally postmenopausal for at least 12 consecutive months (i.e., has had menses at any time in the preceding 12 consecutive months)

Exclusion Criteria:

* Pregnant or breastfeeding
* Severe chronic obstructive pulmonary disease requiring oxygen supplementation
* History of spontaneous pneumothorax
* Active ear/sinus infection. Patients with chronic sinusitis or sinus headaches are excluded unless cleared by ear, nose, throat provider.
* Evidence of pneumothorax or significant pulmonary fibrosis on chest imaging within 60 days of transplant.
* Prior chest surgery requiring thoracotomy or direct chest irradiation.
* Recent of sinus or ear surgery, excluding myringotomy or ear tubes (within the last 5 years).
* Claustrophobia
* Patients who had intrathecal chemotherapy within 2 weeks of starting preparative regimen or cranial irradiation within 4 weeks of starting preparative regimen.
* History of seizures
* No active tobacco use 72 hours prior to transplant until complete transplant recovery.