Using the Anticancer Drug Olaparib to Treat Relapsed/Refractory Acute Myeloid Leukemia or Myelodysplastic Syndrome With an Isocitrate Dehydrogenase (IDH) Mutation

Inclusion Criteria: * Age \>= 18 years. Because no dosing or adverse event (AE) data are currently available on the use of olaparib in patients \< 18 years of age, children are excluded from this study, but will be eligible for future pediatric trials. * Diagnosis of MDS or AML per World Health Organization 2016 classification. AML may be de novo, or following a prior hematologic disorder, including MDS or Philadelphia chromosome-negative myeloproliferative neoplasm, and/or therapy-related. * Patients must have a documented IDH1 or IDH2 mutation within 30 days of inclusion based on mutational testing. Only specific mutations that lead to a neomorphic phenotype will be eligible for enrollment, and include those listed below: * IDH1: R132V, R132G, R132S, R132L, R132C and R132H * IDH2: R140W, R140L, R140Q, R172W, R172G, R172S, R172M, R172K. * Patients with AML or MDS should have disease that has relapsed after, or is refractory to, first-line therapy, with or without subsequent additional therapy. * Patients with MDS should have at least a MDS-excess blasts (EB)1 at the inclusion and have a revised International Prognostic Symptom Score risk stratification of intermediate, high, or very high risk. * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial. * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. * Patients may or may not have been previously treated with IDH targeted therapies. * Patients who have undergone allogeneic stem cell transplant (alloSCT) are eligible if they are \>= 180 days from stem cell infusion, have no evidence of graft versus host disease (GVHD) \> grade 1, and are \>= 2 weeks off all immunosuppressive therapy. * Previous cytotoxic chemotherapy must have been completed at least 3 weeks and radiotherapy at least 2 weeks prior to Day 1 of treatment on the study, and all adverse events (AEs) (excluding alopecia) due to agents administered more than 4 weeks earlier should have recovered to \< grade 1. Patients with hematologic malignancies are expected to have hematologic abnormalities at study entry. Hematologic abnormalities that are thought to be primarily related to leukemia are not considered to be toxicities (AEs) and do not need to resolve to \< grade 1. * Eastern Cooperative Oncology Group (ECOG) performance status 0-2 (Karnofsky \>= 60%). * Patient must have recovered from toxicities of any prior treatment regimen (no Common Terminology Criteria for Adverse Events \[CTCAE\] grading over 1 for non-hematological toxicities, return to baseline for hematological values). * Ability to understand and the willingness to sign a written informed consent document. Patients with impaired decision-making capacity may have a close relative, guardian, caregiver, or legally authorized representative consent on their behalf. * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) unless considered due to Gilbert's syndrome (measured within 28 days prior to administration of study treatment). * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal unless considered due to organ leukemic involvement (measured within 28 days prior to administration of study treatment). If liver metastases are present in which case they must be =\< 5 x ULN. * Creatinine clearance of \> 30 ml/min (measured within 28 days prior to administration of study treatment). * Patients are eligible for this study if low blood count and transfusion support are due to the MDS/AML. * Patients must have, in the best estimate of the treating physician, a life expectance of at least 12-16 weeks. * Postmenopausal or evidence of non-childbearing status for women of childbearing potential: negative urine or serum pregnancy test within 28 days of study treatment and confirmed prior to treatment on day 1. Postmenopausal is defined as: * Amenorrheic for 1 year or more following cessation of exogenous hormonal treatments * Luteinizing hormone (LH) and follicle stimulating hormone (FSH) levels in the post menopausal range for women under 50 * Radiation-induced oophorectomy with last menses \> 1 year ago * Chemotherapy-induced menopause with \> 1 year interval since last menses * Surgical sterilization (bilateral oophorectomy or hysterectomy). * Male patients must use a condom during treatment and for 3 months after the last dose of olaparib when having sexual intercourse with a pregnant woman or with a woman of childbearing potential. Female partners of male patients should also use a highly effective form of contraception if they are of childbearing potential. Exclusion Criteria: * Patients with acute promyelocytic leukemia. * Patients with active central nervous system (CNS) leukemia or requiring maintenance intrathecal chemotherapy. * Patients receiving concurrent chemotherapy, radiation therapy, or immunotherapy for AML/MDS. * Patients actively receiving any other investigational agents. * Management of treatment for patients with co-occurring mutations, like FLT3, will be prioritized by the treating physician after discussion of treatment options with the patient. * Hyperleukocytosis with \> 50,000 white blood cell (WBC)/mcl. Hydroxyurea for WBC count control is permitted before starting treatment and may be continued until day 28 of cycle 1. The maximum dose of hydrea will be 6 grams per day. Patients will be withdrawn from the study if \> 50,000 WBC/mcl occur or recur \> 14 days after starting treatment on the study. * Active, uncontrolled infection. Patients with infection controlled with antibiotics are eligible. * Patients considered a poor medical risk due to a serious, uncontrolled medical disorder, non-malignant systemic disease or active, uncontrolled infection. Examples include, but are not limited to, uncontrolled ventricular arrhythmia, recent (within 3 months) myocardial infarction, uncontrolled major seizure disorder, unstable spinal cord compression, superior vena cava syndrome, extensive interstitial bilateral lung disease on high resolution computed tomography (HRCT) scan or any psychiatric disorder that prohibits obtaining informed consent. * Patients who are pregnant or nursing. Pregnant women are excluded from this study because olaparib is a PARP inhibitor with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for AEs in nursing infants secondary to treatment of the mother with olaparib, breastfeeding should be discontinued if the mother is treated with olaparib. These potential risks may also apply to other agents used in this study. * Resting electrocardiogram indicating uncontrolled, potentially reversible cardiac conditions, as judged by the investigator (e.g., unstable ischemia, uncontrolled symptomatic arrhythmia, congestive heart failure, corrected QT by Fridericia's formula (QTcF) prolongation \> 500 ms, electrolyte disturbances, etc.), or patients with congenital long QT syndrome. * Patients with symptomatic uncontrolled CNS disease. Imaging to confirm the absence of brain metastases is not required. Patients with spinal cord compression unless considered to have received definitive treatment for this and evidence of clinically stable disease for 28 days. * The patient can receive a stable dose of corticosteroids, up to 20 mg by mouth (PO) prednisone daily, before and during the study as long as these were started at least 4 weeks prior to treatment. * Patients unable to swallow orally administered medication and patients with gastrointestinal disorders likely to interfere with absorption of the study medication. * Any previous treatment with PARP inhibitor, including olaparib. * Concomitant use of known strong (e.g., phenobarbital, enzalutamide, phenytoin, rifampicin, rifabutin, rifapentine, carbamazepine, nevirapine and St John's wort) or moderate CYP3A inducers (e.g., bosentan, efavirenz, modafinil). The required washout period prior to starting olaparib is 5 weeks for enzalutamide or phenobarbital and 3 weeks for other agents. * Major surgery within 2 weeks of starting study treatment and patients must have recovered from any effects of any major surgery. * Patients with a known hypersensitivity to olaparib or any of the excipients of the product. * Patient with active malignancies requiring active treatment that interferes with protocol therapy and/or with significant risk of clinical relapse within 12 months that would require treatment interfering with protocol therapy are excluded. * Persistent toxicities (\> Common Terminology Criteria for Adverse Event \[CTCAE\] grade 2) caused by previous cancer therapy, excluding alopecia. * Patients receiving any systemic chemotherapy or radiotherapy (except for palliative reasons) within 3 weeks prior to study treatment. * Concomitant use of known strong CYP3A inhibitors (e.g. itraconazole, telithromycin, clarithromycin, protease inhibitors boosted with ritonavir or cobicistat, indinavir, saquinavir, nelfinavir, boceprevir, telaprevir) or moderate CYP3A inhibitors (e.g. ciprofloxacin, erythromycin, diltiazem, fluconazole, verapamil). The required washout period prior to starting olaparib is 2 weeks. Patients without reasonable alternative may be included in the trial after discussion with the medical monitor. * Previous double umbilical cord blood transplantation (dUCBT). * Breast feeding women.