Sunitinib Malate and Capecitabine in Treating Patients With Unresectable or Metastatic Liver Cancer

Inclusion Criteria:

* Histologically confirmed diagnosis of hepatocellular carcinoma (HCC) OR meets radiographic criteria for diagnosis of HCC without biopsy
* Liver mass at least 1 cm up to 2 cm in size: classic enhancement on 2 approved imaging modalities
* Liver mass \> 2 cm in size: classic enhancement on 1 approved imaging modality
* At least one site of bidimensional measurable disease with the longest axis \>= 20mm by conventional computed tomography (CT) scan or \>= 10mm by spiral CT scan or \>= 10mm by magnetic resonance imaging (MRI)
* Not eligible for curative intent surgery and not eligible for, or not willing to undergo, orthotopic liver transplantation
* Patient has received =\< 1 prior systemic therapy
* Patient has completed treatment with surgery at least 4 weeks prior to study drug administration
* Patient has completed other cancer directed treatments including systemic chemotherapy, transarterial chemotherapy, transarterial chemoembolization or bland embolization, targeted therapy, radiotherapy, or treatment with other investigational anti-cancer agents at least 4 weeks prior to study drug administration AND has radiographic evidence of disease progression following these treatments
* Life expectancy of greater than 12 weeks
* Child-Pugh class A or B
* Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 (Karnofsky \> 60%)
* Platelet count \>= 75,000/mm\^3
* Absolute neutrophil count \>= 1,500/mm\^3
* Alanine aminotransferase (ALT)/aspartate aminotransferase (AST) =\< 5 times upper limit of normal (ULN)
* Total bilirubin =\< 3 times ULN
* Calculated or measured creatinine clearance \>= 40 mL/min
* Prothrombin time =\< 1.5 international normalized ratio (INR)
* Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately
* Ability to understand and the willingness to sign a written informed consent document and comply with scheduled visits, treatment plan, laboratory testing, and other trial procedures

Exclusion Criteria:

* History of another cancer within the last 5 years with the exception of localized basal or squamous cell carcinoma of the skin or stage 1A cervical cancer
* Known brain metastases, spinal cord compression, or evidence of symptomatic brain or leptomeningeal carcinomatosis on screening CT or MRI scan
* National Cancer Institute (NCI)-Common Terminology Criteria for Adverse Events (CTCAE) version 3.0 Grade 2 variceal bleed within 6 weeks of registration or Grade 3 other bleed within 4 weeks of registration
* Any of the following within the 6 months prior to registration: myocardial infarction, severe/unstable angina, coronary/peripheral artery bypass graft, symptomatic congestive heart failure, cerebrovascular accident or transient ischemic attack, or pulmonary embolism
* Ongoing cardiac dysrhythmias of NCI CTCAE Version 3.0 Grade 2
* Prolonged QTc interval on baseline electrocardiograph (EKG)
* Uncontrolled Hypertension (\> 150/100 mm Hg despite optimal medical therapy)
* Severe hepatic impairment, defined as Childs-Pugh Class C
* Pre-existing thyroid abnormality with thyroid function that cannot be maintained in the normal range with medication
* Concurrent treatment on another clinical trial; supportive care trials or non-treatment trials, e.g. quality of life (QOL), are allowed
* Pregnancy or breastfeeding; female subjects must be surgically sterile or be postmenopausal, or must agree to use effective contraception during the period of therapy; all female subjects with reproductive potential must have a negative pregnancy test (serum or urine) prior to enrollment; male subjects must be surgically sterile or must agree to use effective contraception during the period of therapy; the definition of effective contraception will be based on the judgment of the principal investigator or a designated associate
* Other severe acute or chronic medical or psychiatric condition or laboratory abnormality that may increase the risk associated with study participation or study drug administration, or may interfere with the interpretation of study results, and in the judgment of the investigator would make the subject inappropriate for entry into this study
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to sunitinib or capecitabine