Omacetaxine and Venetoclax for the Treatment of Relapsed or Refractory Acute Myeloid Leukemia or Myelodysplastic Syndrome Harboring Mutant RUNX1

Inclusion Criteria: * Patients with a diagnosis of relapsed or refractory acute myeloid leukemia (AML) (or biphenotypic or bilineage leukemia including a myeloid component) or myelodysplastic syndrome * For myelodysplastic syndrome (MDS) patients, patients must have no response, progression, or relapse following at least 4 cycles of azacytidine or decitabine; and/or intolerance defined as grade \>= 3 drug-related toxicity precluding continued therapy * Age \>= 18 years * Subjects must have documented RUNX1 gene mutation * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 * Creatinine \< 2 unless related to the disease * Direct bilirubin \< 2x upper limit of normal (ULN) unless increase is due to Gilbert's disease or leukemic involvement * Aspartate aminotransferase (AST) and/or alanine aminotransferase (ALT) \< 3x ULN unless considered due to leukemic involvement * In the absence of rapidly proliferative disease, the interval from prior treatment to time of initiation will be at least 7 days for cytotoxic or non-cytotoxic (i.e. immunotherapy) agents. Oral hydroxyurea and/or cytarabine (up to 2 g/m\^2) for patients with rapidly proliferative disease is allowed before the start of study therapy, as needed, for clinical benefit and after discussion with the principal investigator (PI) * Male subjects must agree to refrain from unprotected sex and sperm donation from initial study drug administration until 90 days after the last dose of study drug * Willing and able to provide informed consent Exclusion Criteria: * Patients with t(15;17) karyotypic abnormality or acute promyelocytic leukemia (French-American-British \[FAB\] class M3-AML) * Patients with any concurrent uncontrolled clinically significant medical condition including active infection or psychiatric illness, which could place the patient at unacceptable risk of study treatment * Patients with active graft-versus-host-disease (GVHD) status post stem cell transplant (patients without active GVHD on chronic suppressive immunosuppression and/or phototherapy for chronic skin GVHD are permitted after discussion with the PI) * Patients with any severe gastrointestinal or metabolic condition which could interfere with the absorption of oral study medications * Known active hepatitis B virus (HBV) or hepatitis C virus (HCV) infection or known human immunodeficiency virus (HIV) infection * Subject has a white blood cell count \> 25 x 10\^9/L. (Note: Hydroxyurea is permitted to meet this criterion.) * Nursing women, women of childbearing potential (WOCBP) with positive urine pregnancy test, or women of childbearing potential who are not willing to maintain adequate contraception * Appropriate highly effective method(s) of contraception include oral or injectable hormonal birth control, intrauterine device (IUD), and double barrier methods (for example a condom in combination with a spermicide)