Carfilzomib and Melphalan Before Stem Cell Transplant in Treating Patients With Multiple Myeloma

Inclusion Criteria: * Serum creatinine =\< 2 mg/dL * Absolute neutrophil count \>= 1000/uL * Platelet count \>= 50,000/uL * Hemoglobin \>= 8.0 g/dL * Diagnosis of symptomatic MM * Measurable disease of multiple myeloma at the time of baseline values for disease assessment as defined by at least one of the following: * Serum monoclonal protein \>= 1.0 g/dL * \>= 200 mg of monoclonal protein in the urine on 24 hour electrophoresis * Serum immunoglobulin free light chain \>=10 mg/dL AND abnormal serum immunoglobulin kappa to lambda free light chain ratio * Bone marrow plasma cells \>= 30% * NOTE: For patients with no relapse prior to transplant, measurable disease at the time of diagnosis * NOTE: For patients who have had a disease relapse prior to transplant, measurable disease at the time of the most recent relapse immediately prior to transplant; NOTE: If the patient had treatment for the relapsed disease prior to transplant, the patient must have measurable disease at the time of relapse prior to this therapy * Patient is considered for autologous stem cell transplantation with full dose melphalan (200 mg/m\^2) * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2 * Recovered from toxicity of previous chemotherapy (excludes grade 1 neurotoxicity and hematological toxicity) * Provide informed written consent * Ejection fraction \>= 45% * Corrected pulmonary diffusion capacity of greater than or equal to 50% * Forced expiratory volume in 1 second (FEV1) \>= 50% * Forced vital capacity (FVC) \>= 50% * Negative pregnancy test performed =\< 7 days prior to registration, for women of childbearing potential only * Willing to return to Mayo Clinic Rochester, Mayo Clinic Arizona, Mayo Clinic Florida for treatment * Note: During the active monitoring phase of a study (i.e., active treatment and observation), participants must be willing to return to the consenting institution for follow-up * Willing to provide blood and bone marrow samples for correlative research purposes Exclusion Criteria: * Prior autologous or allogeneic bone marrow/peripheral blood stem cell transplant * More than two prior regimens for therapy of MM * Myocardial infarction within 6 months prior to enrollment, or has New York Heart Association (NYHA) class III or IV heart failure, uncontrolled angina, severe uncontrolled ventricular arrhythmias, or electrocardiographic evidence of acute ischemia or active conduction system abnormalities; NOTE: Prior to study entry, any electrocardiogram (ECG) abnormality at screening has to be documented by the investigator as not medically relevant * Seroreactivity for human immunodeficiency virus (HIV), human T-lymphotropic virus (HTLV) I or II, hepatitis B virus (HBV), hepatitis C virus (HCV) * Other active malignancy \< 2 years prior to registration; EXCEPTIONS: Non-melanotic skin cancer or carcinoma-in-situ of the cervix; NOTE: If there is a history or prior malignancy, they must not be receiving other specific treatment for their cancer * Any of the following: * Pregnant women or women of reproductive capability who are unwilling to use effective contraception * Nursing women * Men who are unwilling to use a condom (even if they have undergone a prior vasectomy) while having intercourse with any woman, while taking the drug and for 28 days after stopping treatment * Other co-morbidity, which would interfere with patient's ability to participate in the trial, e.g. uncontrolled infection, uncompensated lung disease * Concurrent chemotherapy, radiotherapy, or any ancillary therapy considered investigational; NOTE: Bisphosphonates are considered to be supportive care rather than therapy, and are thus allowed while on protocol treatment * Known allergies to any of the components of the investigational treatment regimen or required ancillary treatments