• Patients must have histologically confirmed advanced/metastatic or unresectable solid tumor that is refractory to standard therapy or has relapsed after standard therapy

• Patients must have one of the following:

⁃ For Part 1 and 2 -

∙ Triple negative breast cancer (TNBC) (estrogen receptor =\< 1%, progesterone receptor =\< 1%, human epidermal growth factor receptor 2 0-1+ or non-amplified)

‣ Solid tumor with genomic alteration(s) activating RAS signaling including activating KRAS, NRAS, HRAS, or BRAF mutations, inactivating NF1 mutations, or BRAF fusions

• Genomic alterations should be identified locally by next generation sequencing (NGS). Patient genomic reports will be reviewed by the MD Anderson Cancer Center (MDACC) Precision Oncology Decision Support team prior to initiation of study treatment

• For Part 1, patients can have evaluable or measurable disease. For Part 2, patients must have measurable disease by the Response Evaluation Criteria in Solid Tumors (RECIST) v1.1

• Patients must be \>= 4 weeks beyond treatment with any chemotherapy (6 weeks for nitrosoureas or mitomycin C) or other investigational therapy to include hormonal, biological, or targeted agents; or at least 5 half-lives from hormonal, biological, or targeted agents, whichever is shorter at the time of study treatment initiation. Patients must be \>= 4 weeks beyond radiotherapy

• Age \>= 18 years. Because no dosing or adverse events (AE) data are currently available on the use of binimetinib and ZEN003694 (ZEN-3694) in patients \< 18 years of age, children are excluded from this study

• Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%)

• Absolute neutrophil count \>= 1,500/mcL

• Platelets \>= 125,000/mcL

• Hemoglobin \>= 8 g/dL or \>= 5.6 mmol/L

• Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) OR \< 2.0 x ULN in patients with documented Gilbert's syndrome

• Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional ULN

• Calculated creatinine clearance \>= 60 mL/min/1.73 m\^2 (based on the calculated Chronic Kidney Disease-Epidemiology collaboration \[CKD-EPI\] glomerular filtration rate estimation)

• Prothrombin time =\< 1.5 x ULN

• Partial thromboplastin time =\< 1.5 x ULN

• Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this study

• 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 treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression by imaging for at least 4 weeks prior to the first dose of study treatment and any neurologic symptoms have returned to baseline, have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to the first dose of study treatment. This exception does not include carcinomatous meningitis and primary CNS cancers, which are excluded regardless of clinical stability

• 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 study

• Patients should be New York Heart Association Functional Classification of class 2B or better

• Patients must have corrected QT (QTcF) \< 450 msec

• The effects of ZEN003694 (ZEN-3694) and binimetinib on the developing human fetus are unknown. For this reason and because BETi agents are known to be teratogenic, women of childbearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and 4 months after the completion of ZEN003694 (ZEN-3694) and binimetinib administration. 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 ZEN003694 (ZEN-3694) and binimetinib 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