Pembrolizumab, Combination Chemotherapy, and Radiation Therapy Before Surgery in Treating Adult Patients With Locally Advanced Gastroesophageal Junction or Gastric Cardia Cancer That Can Be Removed by Surgery

Inclusion Criteria:

* Histologically or cytologically confirmed adenocarcinoma involving the gastroesophageal junction or gastric cardia
* Central pathology review to determine evaluability of archived esophagogastroduodenoscopy (EGD)/biopsy sample

  * NOTE: If archived sample was collected \> 8 weeks prior to pre-registration (reg), is not available in a timely manner, or was collected outside of Mayo Clinic and considered unevaluable, then baseline EGD with primary tumor biopsy at Mayo Clinic must be performed unless clinically contraindicated; patient is allowed to enroll regardless of whether this Mayo Clinic tissue sample is evaluable; (Only 1 EGD with primary tumor biopsy performed at Mayo Clinic =\< 8 weeks prior to pre-reg is required)
  * NOTE: For both archival or newly obtained tissue, only biopsies are adequate (fine needle aspiration \[FNA\] is not adequate)
* Willing to provide mandatory tissue samples for research purposes
* Baseline imaging with an fludeoxyglucose (FDG)-positron emission tomography (PET) scan negative for distant metastatic disease must be obtained =\< 28 days prior to registration
* Surgically resectable (T2N0, T3N0, Tany with node positivity, M0), as determined by endoscopic ultrasound (EUS) and the following minimum diagnostic work-up:

  * Whole-body PET/computed tomography (CT) (PET/CT of skull base to mid-thigh is acceptable)
  * EUS =\< 21 days prior to registration
  * NOTE: Patients may have regional adenopathy including para-esophageal, gastric, gastrohepatic and celiac nodes; if celiac adenopathy is present, it must be \< 2 cm
  * NOTE: If patient unable to have PET/CT then CT chest/abdomen/pelvis with contrast (preferred) or MRI chest/abdomen/pelvis with contrast
* Surgical consultation at enrolling site to confirm that patient will be able to undergo curative resection after completion of chemoradiation =\< 56 days prior to registration

  * Tumor is amenable to standard resection and reconstruction
* Radiation oncology consultation at enrolling site to confirm that disease can be encompassed in a radiotherapy field =\< 56 days prior to registration

  * NOTE: Radiotherapy quality assurance rapid review must be performed before the first fraction of radiation therapy (RT) is administered; if RT constraints cannot be met, the patient will be removed from the protocol prior to treatment
* Consultation with a medical oncologist at enrolling site =\< 56 days prior to registration, with determination that treatment with neoadjuvant chemoradiotherapy with weekly carboplatin and paclitaxel is considered acceptable
* Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1
* Adequate oral intake and nutritional status without current or likely need for enteral or parenteral feeding during chemoradiation or the preoperative period
* Pre-treatment pulmonary function tests (PFTs), collected =\< 90 days prior to enrollment, must show forced expiratory volume in one second (FEV1) \> 60% of predicted
* Adequate organ function =\< 21 days prior to registration:

  * Aspartate transaminase (AST) level =\< 2.5 x upper limit of normal (ULN) and alanine transaminase (ALT) =\< 3 x upper limit of normal (ULN)
  * Total bilirubin level of =\< 1.5 x ULN
  * Creatinine level =\< 1.2 x ULN or creatinine clearance \> 60 mL/min/1.73 m\^2 for patients with creatinine levels above or below the institutional normal
  * Hemoglobin (Hgb) \>= 9 g/dl without transfusion or epoetin dependency (=\< 7 days prior to assessment)
  * Absolute neutrophil count \>= 1.5 x 10\^9/L
  * Platelets \>= 100 x 10\^9/L
  * Albumin \>= 2.5 g/dl
* Female patients of childbearing potential must be willing to use 2 methods of birth control or be surgically sterile, or abstain from heterosexual activity for the course of the study through 120 days after the last dose of study medication

  * NOTE: Patients of childbearing potential are those who have not been surgically sterilized or have not been free from menses for \> 1 year
* Female patients of childbearing potential must have a negative urine or serum pregnancy test =\< 7 days prior to registration

  * NOTE: If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required
* Male patients must agree to use an adequate method of contraception starting with the first dose of study therapy through 120 days after the last dose of study therapy

  * NOTE: Abstinence is acceptable if this is the established and preferred method of contraception for the subject
* Provide signed written informed consent
* Willing to return to enrolling institution for follow-up
* Willing to provide mandatory tissue and blood samples for research purposes

  * NOTE: Patients must be willing to provide at the time surgical resection; for patients who do not undergo surgery, any on-study tumor biopsy obtained for clinical purposes subsequent to the baseline biopsy must also be available for analysis

Exclusion Criteria:

* Tumor characteristics - any of the following are excluded:

  * Evidence of distant metastases
  * Tumors whose location is restricted to the tubular esophagus (i.e., without involvement of the GEJ or cardia)
  * Tumors whose proximal end are at the level of the carina or higher
  * Invasion of the tracheobronchial tree or presence of tracheoesophageal fistula
  * Palpable supraclavicular nodes, biopsy-proven involvement of supraclavicular nodes, or radiographically involved supraclavicular nodes (\> 1.5 cm in greatest dimension)
  * T1N0M0, T4Nany, or in situ carcinoma
  * Tumor must not extend 5 or more cm into the stomach
* Received prior treatment or receiving current treatment for this malignancy
* Prior radiation to chest or abdomen, or to \> 30% of the marrow cavity
* Inadequate caloric or fluid intake whereby there is a current or likely future need for enteral or parenteral feeding during chemoradiation or the preoperative period
* Major surgery =\< 4 weeks prior to registration
* Active autoimmune disorders, including patients known to be human immunodeficiency virus (HIV) positive, or those requiring chronic steroid administration (excluding inhaled steroids)
* Uncontrolled diabetes (i.e., will interfere with the performance of the FDG PET/CT scans)
* Prior allergic reactions to drugs containing Cremophor, such as teniposide or cyclosporine, or study drugs involved in this protocol, or to a monoclonal antibody or prior hypersensitivity to platinum-containing agents
* Heart conditions - any of the following:

  * Any atrial fibrillation =\< 3 months prior to registration
  * Unstable angina =\< 12 months prior to registration
  * Prior symptomatic congestive heart failure
  * Documented myocardial infarction =\< 6 months prior to registration (pretreatment electrocardiogram \[ECG\] evidence of infarct only will not exclude patients)
  * Prior significant ventricular arrhythmia requiring medication
  * Prior 2nd or 3rd degree heart block or other types of clinically significant conduction delay =\< 6 months prior to registration
  * Clinically significant pericardial disease (including pericardial effusion, pericarditis) or cardiac valvular disease =\< 12 months prior to registration
  * NOTE: As part of history and physical, all patients must be assessed for signs or symptoms of cardiac disease, or for prior history of cardiac disease; these conditions include but are not limited to diseases related to cardiac valves, pericardium, myocardium, atrioventricular delays or arrhythmias; it is strongly recommended that signs or symptoms of potentially clinically significant disease be evaluated with comprehensive cardiac echo
* Prior pancreatitis that was symptomatic or required medical intervention =\< 6 months prior to registration (known toxicity of pembrolizumab)
* Prior enteritis that was symptomatic or required medical intervention =\< 6 months prior to registration (known toxicity of pembrolizumab)
* Uncontrolled hyper/hypothyroidism or hyper/hypocorticism =\< 6 months prior to registration (known toxicity of pembrolizumab)
* Pulmonary conditions - any of the following:

  * Respiratory condition that required any oxygen supplementation =\< 6 months prior to registration
  * Prior or current pneumonitis
  * Clinically significant pulmonary hypertension =\< 12 months prior to registration
  * Lung infection requiring treatment =\< 3 months prior to registration
  * Pulmonary embolism requiring treatment =\< 6 months prior to registration
  * Pleural effusion requiring drainage =\< 12 months prior to registration
* Prior fistula within thorax, including bronchoalveolar or esophageal
* Body mass index (BMI) \>= 35 mg/m\^2 =\< 56 days prior to registration
* Pre-existing motor or sensory neurotoxicity greater than Common Terminology Criteria for Adverse Events (CTCAE) grade 1
* Acute bacterial, viral, or fungal infection requiring treatment at the time of registration
* Active infection or other serious underlying medical condition which would impair the ability of the patient to receive the planned treatment
* Uncontrolled intercurrent illness including, but not limited to, psychiatric illness/social situations, or other co-morbid systemic illnesses or severe concurrent diseases which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens
* Prior malignancy =\< 5 years prior to registration (except non-melanotic skin cancer or carcinoma-in-situ of the cervix) (must be disease free for a minimum of 5 years); if there is a history of prior malignancy, patient must not be receiving other specific treatment (other than hormonal therapy) for cancer
* Dementia or altered mental status that would prohibit the understanding and giving of informed consent
* Any of the following because this study involves an agent where the genotoxic, mutagenic and teratogenic effects are unknown:

  * Pregnant or breastfeeding
  * Patient of childbearing potential who is unwilling to employ adequate contraception
* Received live vaccine =\< 30 days prior to registration