Inclusion Criteria: * DOSE ESCALATION PHASE: * Patients must have histologically confirmed malignancy that is metastatic or unresectable and for which standard curative or palliative measures do not exist or are no longer effective * Patients must have HER2-positive or HER2-expressing tumors determined by a Clinical Laboratory Improvement Act (CLIA)-certified laboratory. Specific requirement of HER2 status is outlined below: * Dose Escalation Module 1 and Module 2: * HER2 1-3+ expression by IHC OR * HER2 amplification by next generation sequencing panel (NGS) or in situ hybridization (ISH) OR * If local testing is not feasible, patients will submit archival tissue for central HER2 testing to determine eligibility. Patients with unknown or negative HER2 testing will not be eligible * Dose Escalation Module 3: * HER2 1-2+ expression by IHC OR * HER2 amplification by next generation sequencing panel (NGS) or in situ hybridization (ISH) OR * If local testing is not feasible, patients will submit archival tissue for central HER2 testing to determine eligibility. Patients with unknown or negative HER2 testing will not be eligible * DOSE EXPANSION PHASE: * Patients must have histologically confirmed platinum resistant, high grade serous ovarian carcinoma. Platinum resistant is defined as radiographic progression \< 6 months following the last dose of platinum therapy * HER2 IHC 1+ per local or central testing * There must be least one lesion suitable for biopsy without significant risk to the patient * Patient disease must be evaluable or measurable by Response Evaluation Criteria in Solid Tumors (RECIST) version 1.1. Biopsiable lesion cannot be the only RECIST-measurable lesion * DOSE ESCALATION AND DOSE EXPANSION PHASES: * Patients must have had at least one prior line of cytotoxic chemotherapy * Patients can have received an unlimited number of additional lines of chemotherapy, targeted therapy, biologic therapy, or hormonal therapy * Patients must have archival formalin-fixed paraffin-embedded (FFPE) tissue available for central confirmation of HER2 testing * Age \>= 18 years. Because no dosing or adverse event data are currently available on the use of DS-8201a in combination with olaparib in patients \< 18 years of age, children are excluded from this study * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 (Karnofsky \>= 70%) * Hemoglobin \>= 10.0 g/dL (within 21 days of randomization/enrollment) * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * Absolute neutrophil count \>= 1,000/mcL (within 21 days of randomization/enrollment) * No administration of granulocyte colony-stimulating factor (G-CSF) is allowed within 1 week prior to screening assessment * Platelets \>= 100,000/mcL (within 21 days of randomization/enrollment) * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN), (\< 3 x ULN in the presence of documented Gilbert's syndrome or liver metastases at baseline) (within 21 days of randomization/enrollment) * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 x institutional ULN (within 21 days of randomization/enrollment) * Creatinine =\< 1.5 x institutional ULN OR glomerular filtration rate (GFR) \>= 51 mL/min/1.73 m\^2 unless data exists supporting safe use at lower kidney function values, no lower than 30 mL/min/1.73 m\^2 (using the Cockcroft-Gault Equation) (within 21 days of randomization/enrollment) * Patients must have left ventricular ejection fraction (LVEF) \>= 50% by either an echocardiogram (ECHO) or multigated acquisition (MUGA) scan within 28 days before randomization/enrollment * Patients who are human immunodeficiency virus (HIV) positive may participate IF they meet the following eligibility requirements: * They must be stable on their anti-retroviral regimen, and they must be healthy from an HIV perspective * They must have an undetectable viral load and a CD4 count \>= 250 cells/uL within 7 days of enrollment * They must not be currently receiving prophylactic therapy for an opportunistic infection and must not have had an opportunistic infection within the past 6 months * HIV-infected patients should be monitored every 12 weeks for viral load and CD4 counts * 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 with brain metastases should be stable and off steroids and at least 4 weeks from radiation at the time of registration * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be better than class 2B * The effects of DS-8201a and olaparib on the developing human fetus are unknown. For this reason and because HER2 antibody conjugated to a topoisomerase 1 inhibitor agents as well as PARP inhibitors are known to be teratogenic; thus, women of child-bearing potential and men must agree to use highly effective contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for at least 7 months (women of childbearing potential \[WOCBP\] only) after the last dose of study drug. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of DS-8201a and olaparib administration * Women of non-child-bearing potential defined as pre-menopausal females with a documented tubal ligation or hysterectomy; or postmenopausal defined as 12 months of spontaneous amenorrhea (in questionable cases, a blood sample with simultaneous follicle-stimulating hormone \[FSH\] \> 40 mIU/mL and estradiol \< 40 pg/mL \[\< 147 pmol/L\] is confirmatory) are eligible. Females on hormone replacement therapy (HRT) and whose menopausal status is in doubt will be required to use one of the contraception methods outlined for women of child-bearing potential if they wish to continue their HRT during the study. Otherwise, they must discontinue HRT to allow confirmation of post-menopausal status prior to study enrollment. For most forms of HRT, at least 2-4 weeks will elapse between the cessation of therapy and the blood draw; this interval depends on the type and dosage of HRT. Following confirmation of their post-menopausal status, they can resume use of HRT during the study without use of a contraceptive method * Male subjects must not freeze or donate sperm starting at screening and throughout the study period, and at least 4 months after the final study drug administration. Preservation of sperm should be considered prior to enrollment in this study * Female subjects must not donate, or retrieve for their own use, ova from the time of screening and throughout the study treatment period, and for at least 7 months after the final study drug administration * Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available will also be eligible Exclusion Criteria: * Patients who have had chemotherapy (including antibody drug therapy) within 4 weeks with the following exceptions: 1 week for weekly paclitaxel; 2 weeks or five half-lives, whichever is longer, for small-molecule targeted agents such as 5-fluorouracil-based agents, folinate agents, hormonal agents; or 6 weeks for nitrosoureas or mitomycin C * Patients who have had radiation therapy within 4 weeks * Patients who have had a major surgery within 4 weeks * Patients who are receiving any other investigational agents * Patients with a history of (non-infectious) interstitial lung disease (ILD)/pneumonitis that required steroids, has current ILD/pneumonitis, or where suspected ILD/pneumonitis cannot be ruled out by imaging at screening * Patients with clinically severe pulmonary compromise resulting from intercurrent pulmonary illnesses including, but not limited to, any underlying pulmonary disorder (i.e. pulmonary emboli within three months of the study enrollment, severe asthma, severe chronic obstructive pulmonary disease \[COPD\], restrictive lung disease, pleural effusion, etc.), and any autoimmune, connective tissue or inflammatory disorders with potential pulmonary involvement (i.e. rheumatoid arthritis, Sjogren's, sarcoidosis, etc.), or prior pneumonectomy * History of allergic reactions attributed to compounds of similar chemical or biologic composition to DS-8201a, the inactive ingredients in the drug product, olaparib, or severe hypersensitivity to other monoclonal antibodies * Patients receiving any medications or substances that are moderate or strong inhibitors or inducers of CYP3A are ineligible. Because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated medical reference. As part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product * Patients with a medical history of myocardial infarction within 6 months before randomization/enrollment, symptomatic congestive heart failure (CHF) (New York Heart Association class IIb to IV), troponin levels consistent with myocardial infarction as defined according to the manufacturer 28 days prior to randomization * Patients with a corrected QT interval (QTc) prolongation to \> 470 ms (females) or \> 450 ms (males) based on average of the screening triplicate 12-lead electrocardiogram (ECG) * Patients with multiple primary malignancies within 3 years, except adequately resected non-melanoma skin cancer, curatively treated in-situ disease, and other solid tumors curatively treated * Patients with an uncontrolled infection requiring IV antibiotics, antivirals, or antifungals * Patients receiving chloroquine or hydroxychloroquine will require a washout period of \>= 14 days to be eligible for the study * Patients with unresolved toxicities from previous anticancer therapy, defined as toxicities (other than alopecia) not yet resolved to grade =\< 1 or baseline. Subjects with chronic grade 2 toxicities may be eligible per the discretion of the investigator after consultation with the sponsor medical monitor or designee (e.g., grade 2 chemotherapy-induced neuropathy) * Patients with psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study because DS-8201a is a HER2 antibody conjugated to a topoisomerase 1 inhibitor agent with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with DS-8201a, breastfeeding should be discontinued if the mother is treated with DS-8201a. These potential risks may also apply to other agents used in this study * Patients who have received a live, attenuated vaccine within 30 days prior to the first dose of DS-8201a

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