Inclusion Criteria:

          -  Participants must have histologically or cytologically confirmed adenocarcinoma of the
             prostate

          -  Participants must have castrate levels of serum testosterone < 50 ng/dL

          -  Participants without orchiectomy must be maintained on luteinizing hormone releasing
             hormone (LHRH) agonist/antagonist; participants receiving prior docetaxel abiraterone,
             or next generation AR antagonist (enzalutamide, apalutamide, or darolutamide) for
             hormone sensitive disease are permitted

          -  Participants must have progressive disease as defined by any of the following:

               -  Castrate resistant disease as defined by PCWG-3 criteria; participants must have
                  a rise in PSA on two successive determination at least one week apart and PSA
                  levels >= 2 ng/mL (only the screening PS needs to be >= 2 ng/mL) and serum
                  testosterone < 50 ng/dL

               -  Soft tissue progression as defined by RECIST version 1.1

               -  Bone disease progression as defined by PCWG-3 criteria including the development
                  of two or more new lesions on bone scan

          -  Participants must have >= 2 bone metastases by radiographic imaging and at least 1
             lesion which has not been treated with prior radiation therapy

          -  Participants must have tumor accessible for biopsy and be agreeable to baseline tumor
             biopsy; a metastatic focus is preferred but if not available and prostate is still
             intact prostate biopsy can be performed

          -  Availability at the study site of formalin-fixed, paraffin-embedded (FFPE) archival
             tumor specimens, when available

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 1 (Karnofsky >= 80%)

          -  White blood cell count (WBC) >= 3,000/mcL (within 28 days prior to administration of
             study treatment)

          -  Absolute neutrophil count (ANC) >= 1,500/mcL (within 28 days prior to administration
             of study treatment)

          -  Platelets >= 100,000/mcL (within 28 days prior to administration of study treatment)

          -  Hemoglobin >= 10 g/dL (transfusions permitted) (within 28 days prior to administration
             of study treatment)

          -  Total bilirubin =< 1.5 x the institutional upper limit of normal (ULN) (within 28 days
             prior to administration of study treatment); for subjects with Gilbert's disease =<
             3.0 mg/dL

          -  Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =< 3 x institutional
             ULN (within 28 days prior to administration of study treatment)

          -  Creatinine clearance >= 51 ml/min as defined by Cockcroft-Gault equation (within 28
             days prior to administration of study treatment)

          -  Participants should be receiving an osteoclast targeting agent including either
             bisphosphonates or denosumab except in patients with contraindications as determined
             by the treating investigator including:

               -  Hypocalcemia

               -  Hypophosphatemia

               -  Renal impairment including those with a glomerular filtration rate < 35 mL/min
                  using the Cockcroft-Gault equation

               -  Hypersensitivity to drug formulation

               -  Dental condition or need for dental intervention that per the investigator would
                  increase the risk of osteonecrosis of the jaw

          -  The effects of olaparib and radium-223 on the developing human fetus are unknown; for
             this reason, men treated or enrolled on this protocol must agree to use adequate
             contraception and avoid sperm donation prior to the study, for the duration of study
             participation, and three months after discontinuation of olaparib and radium-223
             administration

          -  Human immunodeficiency virus (HIV)-positive with negative viral loads on stable
             antiretroviral regimen and CD4 count > 250 are eligible

          -  Ability to understand and the willingness to sign a written informed consent document;
             patients with impaired decision-making who have a legal guardian (e.g., spouse) able
             to make informed decisions on behalf of the patient are eligible

          -  Patients must be able to tolerate oral medications by mouth and not have a
             gastrointestinal illness that would preclude absorption of olaparib

        Exclusion Criteria:

          -  Pathology consistent with small cell carcinoma of the prostate

          -  Presence of visceral metastases (liver, lung, brain, etc.) or malignant
             lymphadenopathy exceeding 4 centimeters (cm) in short diameter

          -  Prior treatment with radium-223

          -  Prior treatment with olaparib or other PARPi

          -  Treatment with abiraterone, apalutamide, or darolutamide within 2 weeks of treatment
             initiation; treatment with cytotoxic chemotherapy within 3 weeks of treatment
             initiation; treatment with investigational prostate cancer directed therapy within 4
             weeks of treatment initiation; treatment with enzalutamide within 4 weeks of treatment
             initiation

          -  Prior hemibody external radiotherapy

          -  Palliative radiation therapy to the bone or other sites within 2 weeks of treatment
             initiation

          -  Participants who are receiving any other investigational agents

          -  Imminent or established spinal cord compression based on clinical and/or imaging
             findings

          -  Uncontrolled intercurrent illness including, but not limited to, ongoing or active
             infection requiring need for intravenous anti-microbials, symptomatic congestive heart
             failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social
             situations that would limit compliance with study requirements

          -  Clinically significant medical condition defined as:

               -  Cerebral infarction within 6 months of study treatment

               -  Transient ischemic attack within 3 months of study treatment

               -  Myocardial infarction within 6 months of study treatment

               -  Uncontrolled angina within 3 months of study treatment

               -  Congestive heart failure New York Heart Association (NYHA) class 3 or 4, or
                  subjects with history of congestive heart failure NYHA class 3 or 4 in the past,
                  or history of anthracycline or anthracenedione (mitoxantrone) treatment, unless a
                  screening echocardiogram or multi-gated acquisition scan performed within 3
                  months of the screening visit results in a left ventricular ejection fraction
                  that is >= 45%

               -  History of clinically significant ventricular arrhythmias (e.g., ventricular
                  tachycardia, ventricular fibrillation, torsade de pointes)

               -  Prolonged corrected QT interval by the Fridericia correction formula on the
                  screening electrocardiogram (ECG) > 470 msec (as determined on 2 or more time
                  points within a 24 hour period if the first ECG demonstrates a prolonged
                  corrected QT interval) or family history of long QT syndrome

               -  History of Mobitz II second degree or third degree heart block without a
                  permanent pacemaker in place

               -  Uncontrolled hypertension as indicated by a resting systolic blood pressure > 170
                  mmHg or diastolic blood pressure > 105 mmHg at the screening visit

               -  History of hypertensive emergency or encephalopathy within 6 months of study
                  treatment

               -  Deep venous thrombosis or pulmonary embolism within 3 months of study treatment

          -  Major surgery within 4 weeks of study treatment; subjects with clinically relevant
             ongoing complications from prior surgery are not eligible

          -  History of gastrointestinal disorders (medical disorders or extensive surgery) which
             may interfere with the absorption of the study drug

          -  Patient unable to swallow orally administered medication

          -  History of bowel obstruction within 1 month of study treatment

          -  History of abdominal fistula, intra-abdominal abscess, or gastrointestinal perforation
             within the 3 months of study treatment

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to olaparib or radium-223

          -  Participants receiving strong CYP3A4/5 inducers or inhibitors are ineligible;
             dihydropyridine calcium-channel blockers are permitted for management of hypertension;
             the required washout period prior to starting olaparib is 2 weeks for CYP3A
             inhibitors; the required washout period prior to starting olaparib is 4 weeks for
             enzalutamide or phenobarbital and 3 weeks for other CYP3A inducers

          -  Patients with known active hepatitis (i.e. hepatitis B or C) infection

          -  Patients with myelodysplastic syndrome (MDS)/acute myeloid leukemia (AML) or with
             features suggestive of MDS/AML

          -  Patient having received prior allogenic bone marrow transplant or double umbilical
             cord blood transplantation

          -  Individuals with a history of a different malignancy are ineligible except for the
             following circumstances:

               -  Individuals with a history of other malignancies are eligible if they have been
                  disease-free for at least 3 years and are deemed by the investigator to be at low
                  risk for recurrence of that malignancy, or

               -  Individuals with the following cancers are eligible if diagnosed and treated
                  within the past 5 years: superficial bladder cancer, basal cell or squamous cell
                  carcinoma of the skin