Inclusion Criteria:  Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 1. Individuals with the following pathologically-confirmed (cytological diagnosis is adequate) select locally advanced or metastatic solid tumors, whose disease have progressed on or are intolerant to all standard therapy: gastroesophageal (esophageal, gastroesophageal junction, and gastric) adenocarcinoma, pancreatic adenocarcinoma, hepatocellular carcinoma (HCC), nonsquamous non-small cell lung cancer (NSCLC), squamous cell carcinoma of the head and neck (SCCHN), mesothelioma, and triple-negative breast cancer (TNBC). Intolerant participants are those who have developed clinical or laboratory abnormalities that prevent continued drug administration as evaluated by the principal investigator at the time of screening. Must have at least 1 RECIST version 1.1 evaluable lesion.  Adequate bone marrow, renal, and hepatic functions, as determined by the following laboratory parameters:  Absolute neutrophil count (ANC) greater than or equal to (>=) 1000/microliter (mcL), platelet count >=75,000/mcL, and hemoglobin >= 8.0 grams per deciliter (g/dL) without growth factor support for ANC or transfusion support for platelets within 14 days before the first study treatment dose. Total bilirubin <=1.5 times the institutional upper limit of normal (ULN). For participants with Gilbert's disease or HCC, <=3 milligrams per deciliter (mg/dL). Serum alanine aminotransferase and aspartate aminotransferase <=3.0*ULN or <=5.0*ULN with liver metastases or HCC. Albumin >=3.0 g/dL. Calculated creatinine clearance using the Cockcroft-Gault formula >=30 mL/minute. Left ventricular ejection fraction (LVEF) >50%, as measured by echocardiogram or multiple gated acquisition scan (MUGA) within 4 weeks before receiving the first dose of study drug. For participants with HCC only: Child-Pugh score less than or equal to 7 (Child-Pugh A or B7). Clinically significant toxic effects of previous therapy have recovered to Grade 1 (per National Cancer Institute Common Terminology Criteria for Adverse Events [NCI CTCAE] Version 5.0) or baseline, except for alopecia, Grade 2 peripheral neuropathy, and/or autoimmune endocrinopathies with stable endocrine replacement therapy. Previously treated with fully human/humanized antineoplastic monoclonal antibodies must not have received treatment with such antibodies for at least 4 weeks or the time period equal to the dosing interval, whichever is shorter. No washout period is required for prior treatment with pembrolizumab or other anti-programmed cell death protein 1 (PD-1) antibodies, although the first study dose of these drugs must not occur at an interval less than standard of care (that is, 3 weeks for 200 mg of IV pembrolizumab).  Exclusion Criteria:  History of any of the following <=6 months before first dose of study drug(s): congestive heart failure New York Heart Association Grade III or IV, unstable angina, myocardial infarction, persistent hypertension >=160/100 millimeters of mercury (mmHg) despite optimal medical therapy, ongoing cardiac arrhythmias of Grade >2 (including atrial flutter/fibrillation or intermittent ventricular tachycardia), other ongoing serious cardiac conditions (example, Grade 3 pericardial effusion or Grade 3 restrictive cardiomyopathy), or symptomatic cerebrovascular events. Chronic, stable atrial fibrillation on stable anticoagulation therapy, including low molecular-weight heparin, is allowed. QT interval with Fridericia correction method >450 milliseconds (men) or >475 milliseconds (women) on a 12- lead ECG during the screening period. Grade >=2 hypotension (that is, hypotension for which nonurgent intervention is required) at screening or during C1D1 predose assessment. Oxygen saturation <92% on room air at screening or during C1D1 predose assessment. Treated with other STING agonists/antagonists, Toll-like receptor agonists or CCR2 agonist/antagonist within the past 6 months. Active vaping within 90 days of C1D1 of study drug(s). Active smoking. Active diagnosis of pneumonitis, interstitial lung disease, severe chronic obstructive pulmonary disease, idiopathic pulmonary fibrosis, other restrictive lung diseases, acute pulmonary embolism, or Grade >=2 pleural effusion not controlled by tap or requiring indwelling catheters. Grade >=2 fever of malignant origin. Chronic, active hepatitis (example, participants with known hepatitis B surface antigen seropositive and/or detectable hepatitis C virus [HCV] RNA). History of hepatic encephalopathy. Prior or current clinically significant ascites, as measured by physical examination, that requires active paracentesis for control. Treatment with any investigational products or other anticancer therapy (including chemotherapy, targeted agents, and immunotherapy), within 14 days or 5 half-lives, whichever is shorter, before C1D1 of study drug(s). Radiation therapy within 14 days (42 days for radiation to the lungs) and/or systemic treatment with radionuclides within 42 days before C1D1 of study drug(s). Participants with clinically relevant ongoing pulmonary complications from prior radiation therapy are not eligible.  Use of systemic corticosteroids or other immunosuppressive therapy, concurrently or within 14 days of C1D1 of study drug(s), with the following exceptions:  Topical, intranasal, inhaled, ocular, and/or intra-articular corticosteroids. Physiological doses of replacement steroid therapy (example, for adrenal insufficiency), not to exceed the equivalent of 10 mg prednisone daily. Recipients of allogeneic or autologous stem cell transplantation or organ transplantation.  Additional criteria specific for participants in TAK-500 and pembrolizumab combination arm only:  Contraindication to the administration of a pembrolizumab or prior intolerance to pembrolizumab or other anti-PD-1 or anti-programmed cell death-ligand 1 antibody. History of intolerance to any component of the trial treatment agents or known serious or severe hypersensitivity reaction to any of the study drugs or their excipients. (Pembrolizumab is formulated with L-histidine, polysorbate 80, and sucrose,