Chemoembolization With or Without Sorafenib Tosylate in Treating Patients With Liver Cancer That Cannot Be Removed by Surgery

Inclusion Criteria: * Patients must have a diagnosis of hepatocellular carcinoma by at least one criterion listed below: * Histologically confirmed * Magnetic resonance imaging (MRI) or computerized tomography (CT) consistent with liver cirrhosis AND at least one solid liver lesion \> 2 cm with early enhancement and delayed enhancement washout regardless of alpha-feto protein levels (AFP) * AFP \> 400 ng/mL AND evidence of at least one solid liver lesion \> 2 cm regardless of specific imaging characteristics on CT or MRI * Patients must have hepatocellular carcinoma (HCC) limited to the liver; there must be no clinical or radiographic evidence of extrahepatic HCC * Portal lymphadenopathy IS permitted for patients with hepatitis B virus (HBV) or hepatitis C virus (HCV) - as lymphadenopathy is commonly associated with hepatitis unrelated to malignancy * Staging CT of the chest and CT or MRI of the abdomen and pelvis must have been completed within 4 weeks of study registration * Patients must have measurable disease constituting \< 50% of liver parenchyma within 4 weeks of registration * Patients may not have ascites detectable on physical examination * Patients must not be candidates for curative resection, orthotopic liver transplantation, or radiofrequency ablation (RFA) * Patients may have been treated with RFA in the past, but no sooner than 4 weeks before study registration * Patients may have undergone previously attempted curative liver resection * Patients may NOT have been previously treated with brachytherapy such as yttrium-90 microsphere * Patients may NOT have been previously treated with sorafenib, chemoembolization, or systemic chemotherapy including cytotoxic agents or molecularly targeted agents * Branch portal vein invasion by tumor is permitted but patients with main portal vein invasion by tumor are not eligible * Patients must have Child-Pugh score of A or B7 within 4 weeks prior to study registration * Serum total bilirubin =\< 2.0 mg/dL * Alkaline phosphatase \< 5 x upper limit of normal (ULN) * Aspartate aminotransferase (AST), alanine aminotransferase (ALT) \< 5 x ULN * Serum creatinine =\< 1.5 mg/dL * Platelet count \>= 50,000/mm\^3 * Patients must not have any evidence of bleeding diathesis or active gastrointestinal bleeding * Patients must have no clinical signs of heart failure and meet New York Heart Association functional classification I or II defined as: * Class I - patients with no limitation of activities; they suffer no symptoms from ordinary activities * Class II - patients with slight, mild limitation of activity; they are comfortable with rest or with mild exertion * Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Patients must have a life expectancy of at least 3 months * Patients must not be known to be human immunodeficiency virus (HIV) positive * Patients must not have other uncontrolled intercurrent illnesses excluding HBV or HCV, including, but not limited to: uncontrolled hypertension, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia or psychiatric illness/addictive disorders that would limit compliance with study requirements * Uncontrolled hypertension is defined as optimally treated baseline blood pressure that exceeds 150/90 mm Hg * Patients must not be taking cytochrome P450 enzyme inducing drugs * Age \>= 18 years * Women must not be pregnant or breast-feeding; all females of childbearing potential must have a blood test or urine study within 2 weeks prior to registration to rule out pregnancy * Women of childbearing potential and sexually active males must be strongly advised to use an accepted and effective method of contraception * Patients must not have an allergy to iodine or gadolinium contrast that cannot be safely controlled with premedication * Patient must be able to swallow pills, as study medications cannot be crushed