Bortezomib in Treating Patients With Hepatocellular Carcinoma (Liver Cancer)

Inclusion Criteria: * Histologically or cytologically confirmed hepatocellular carcinoma (HCC) that is not amenable to surgical resection * Must have measurable disease; NOTE: For patients having only lesions measuring \> 1 cm to =\< 2 cm must use spiral CT imaging for all tumor assessments * Absolute neutrophil count (ANC) \>= 1500/mm\^3 * PLT \>= 75,000/mm\^3 * Total bilirubin =\< 3 x upper normal limit (UNL) * Serum AST =\< 5 x UNL * Serum ALT =\< 5 x UNL * Serum creatinine =\< 2 mg/dL * Serum albumin \>= 2.5 g/dL * PT/ INR =\< 1.5 (EXCEPTION - Patients with full-dose anticoagulants are eligible provided the patient has been on a stable dose, \>= 2 weeks, of warfarin or low molecular weight heparin and has an PT/INR range 2-3) * Child-Pugh classification of A or B * Patients may not have received prior systemic chemotherapy BUT may have received prior chemoembolization, cryotherapy, radiofrequency ablation, ethanol injection, or photodynamic therapy, provided the following criteria are met: * \> 6 weeks has elapsed since that therapy * Indicator lesion(s) is/are outside the area of prior treatment or, if the only indicator lesion is inside the prior treatment area, there must be clear evidence of disease progression associated with that lesion * Edges of the indicator lesion are clearly distinct on CT scanning * ECOG performance status (PS) 0, 1, or 2 * Estimated life expectancy \>= 24 weeks * Capable of understanding the investigational nature, potential risks and benefits of the study and able to provide written informed consent Exclusion Criteria: * Any of the following: * Prior systemic anticancer therapy. Note: Chemoembolization is allowed and for trial purposes is not considered a systemic chemotherapy; however, \>= 6 weeks must have elapsed between chemoembolization and enrollment on this study * Prior PS-341 therapy * Immunotherapy =\< 4 weeks have elapsed prior to study entry * Biologic therapy =\< 4 weeks have elapsed prior to study entry * Radiation therapy =\< 4 weeks have elapsed prior to study entry * Cryotherapy =\< 6 weeks have elapsed since prior to study entry * Radiofrequency ablation =\< 6 weeks have elapsed since prior to study entry * Ethanol injection =\< 6 weeks have elapsed since prior to study entry * Photodynamic therapy =\< 6 weeks have elapsed since prior to study entry * Major surgery, or significant traumatic injury =\< 3 weeks prior to study entry * Other concurrent chemotherapy, immunotherapy, radiotherapy, or any other therapy or supportive care considered investigational * Presence of \> grade 1 sensory peripheral neuropathy of any etiology OR grade 1 with neuropathic pain of any etiology * History of allergic reactions attributed to compounds of similar chemical or biologic composition to PS-341 * History of other malignancy =\< 3 years prior to study entry, except for adequately treated basal cell or squamous cell skin cancer * Any of the following as this regimen may be harmful to a developing fetus or nursing child: * Pregnant women * Breastfeeding women * Men or women of childbearing potential or their sexual partners who are unwilling to employ adequate contraception (condoms, diaphragm, birth control pills, injections, intrauterine device \[IUD\], surgical sterilization, subcutaneous implants, or abstinence, etc.) * Known CNS metastases * Uncontrolled intercurrent illness including, but not limited to: * Ongoing or active infection * Symptomatic congestive heart failure * Unstable angina pectoris, cardiac arrhythmia * Psychiatric illness that would limit compliance with study requirements * HIV-positive patients receiving combination anti-retroviral therapy