A Study of Pembrolizumab Plus Local Chemotherapy Using Isolated Limb Infusion (ILI) for Patients With Sarcoma in the Arm or Leg

Inclusion Criteria: Patients must fulfill all of the following criteria to be eligible for admission to the study. Any exceptions from the protocol-specific selection criteria must be approved by the Principal Investigator and/or the Institutional Review Board (IRB) before enrollment. * Age \>/= 12 years at the time of informed consent * Willing and able to provide written informed consent/assent for the trial * Willing to comply with treatment protocol * Have a histologically confirmed metastatic and/or locally advanced sarcoma * Eligible for standard treatment with pembrolizumab * Eligible for an isolated limb infusion (ILI) as determined by the treating physician * Have undergone at least one prior line of systemic therapy (e.g. chemotherapy, immunotherapy, targeted or biological therapy) or have declined the standard of care systemic option. * Have measurable disease (at least one index lesion) as defined by RECIST 1.1 or by clinical measurement for superficial lesions not amenable to radiographic surveillance. Index lesions must not be chosen from a previously irradiated field unless there has been radiographically and/or pathologically documented tumor progression in that lesion prior to enrollment. * Adequate performance status: ECOG \</= 2 or KPS \>/= 60% * Adequate organ function determined within 3 weeks of treatment initiation, defined as follows: * Hemoglobin \>/= 8.0 g/dL * Absolute neutrophil count \>/= 1,000/mm\^3 (1.0 x 10\^9/L) * Platelet count \>/= 50,000/mm\^3 (50 x 10\^9/L) * Serum bilirubin \</= 1.5 x upper limit of normal (ULN) OR direct bilirubin \</= ° ° ULN for a patient with total bilirubin level \> 1.5 x ULN Aspartate aminotransferase (AST) \</= 2.5 x ULN OR \</= 5 x ULN for patients with liver metastases * Alanine aminotransferase (ALT) \</= 2.5 x ULN OR \</= 5 x ULN for patients with liver metastases * Alkaline phosphatase \< 5 x ULN * Serum creatinine \</= 1.5 x ULN or a measured or calculated creatinine clearance \>/= 60 mL/min for a patient with creatinine levels \> 1.5 x institutional ULN (Note: Creatinine clearance need not be determined if the baseline serum creatinine is within normal limits. GFR can also be used in place of creatinine or CrCl) * International normalized ratio (INR) or prothrombin time (PT) \</= 1.5 X ULN unless patient is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants * Activated partial thromboplastin time (aPTT) \</= 1.5 x ULN unless subject is receiving anticoagulant therapy as long as PT and PTT is within therapeutic range of intended use of anticoagulants Creatinine clearance should be calculated per institutional standard. For female patients of childbearing potential, negative serum pregnancy test at screening visit and within 72 h prior to the first dose of study medication. Exclusion Criteria: Patients who fulfil any of the following criteria are not eligible for admission to the study: * Have any other malignancy that requires active treatment * Ineligible for ILI because of underlying physical conditions (e.g. coronary artery disease with inability to tolerate anesthesia) as determined by treating physician * Has previously experienced hypersensitivity to pembrolizumab or any of its excipients * Has uncontrolled intercurrent illness including active infection requiring systemic therapy or symptomatic congestive heart failure within the past 6 months * Has known active central nervous system (CNS) metastases. Patients with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least 4 weeks prior to study Day 1 and return to baseline of neurologic symptoms), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to trial treatment. This exception does not include sarcomatous meningitis, which is excluded regardless of clinical stability. * Shows evidence of clinically significant immunosuppression such as the following: * Primary immunodeficiency state such as Severe Combined Immunodeficiency Disease * Concurrent opportunistic infection * Receiving systemic immunosuppressive therapy (\> 2 weeks) including oral steroid doses \> 10 mg/day of prednisone or equivalent within 7 days prior to enrollment. However, in the setting of non-immune mediated indications for use, chronic/active low dose steroid use may be permitted at the discretion of the principal investigator. * Has a known active or chronic infection with HIV if CD4 count is less than 500. * Has a known active infection with hepatitis B or hepatitis C * Has a known history of active tuberculosis infection * Has history or evidence of symptomatic autoimmune disease (e.g., pneumonitis, glomerulonephritis, vasculitis, or other), or history of active autoimmune disease that has required systemic treatment (i.e., use of corticosteroids, immunosuppressive drugs or biological agents used for treatment of autoimmune diseases) in the past 2 years. Replacement therapy (e.g., thyroxine for hypothyroidism, insulin for diabetes or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency) is not considered a form of systemic treatment for autoimmune disease. * For female subjects, is pregnant or breast-feeding, or planning to become pregnant * For male subjects, is planning to father a child within the projected duration of the trial, starting with the pre-screening or screening visit, during study treatment and through 4 months after the last dose of pembrolizumab * For patients of childbearing potential, is unwilling to use acceptable method(s) of effective contraception during study treatment and through 4 months after the last dose of pembrolizumab. (Women not of childbearing potential are defined as: post-menopausal \[age \> 55 years with cessation of menses for 12 or more months or less than 55 years but not spontaneous mens