Study Evaluating the Safety and Efficacy of Eribulin Mesilate in Combination With Irinotecan Hydrochloride in Children With Refractory or Recurrent Solid Tumors

Participants must be * \>=12 months to less than or equal to (\<=) 25 years old at the time of consent \[no more than 25 percent (%) of participants between the ages of 18 and 25 years will be enrolled in this study\]. * In Phase 1, \>6 months and \<12 months at the time of consent (Schedule A only) participants will be enrolled one dose level behind the \>=12 months participant in order to maximize safety for infant participants. In Phase 2, participants aged \>6 months and \<12 months at the time of consent will be enrolled to Schedule A with a modified dose of eribulin with the irinotecan dose maintained in order to maximize safety for infant participants. Inclusion Criteria: * Phase 1: Participants must be diagnosed with histologically confirmed solid tumors (excluding CNS tumors), which is relapsed or refractory, and for which there are no currently available therapies. * Phase 2: Participants must be diagnosed with histologically confirmed RMS, NRSTS or EWS which is relapsed or refractory having received at least 1 prior therapy, including primary treatment. * Phase 1: Participants must have either measurable or evaluable disease as per RECIST 1.1. * Phase 2: Participants must have measurable disease as per RECIST 1.1. * Participant's current disease state must be one for which there is no known curative therapy. * Participant's performance score must be \>=50% Karnofsky (for participants \>16 years of age) or Lansky (for participants \<=16 years of age).Participants who are unable to walk because of paralysis and/or previous surgeries, but who are in a wheelchair, will be considered ambulatory for the purpose of assessing the performance score. * Participants must have fully recovered from the acute toxic effects of all prior anticancer treatments prior to study drug administration: * Must not have received myelosuppressive chemotherapy within 21 days prior to study drug administration (42 days if prior nitrosourea). * Must not have received a long-acting growth factor (example, Neulasta) within 14 days or a short-acting growth factor within 7 days. * Must not have received an antineoplastic targeted therapy within 14 days. * Must not have received immunotherapy, example, tumor vaccines, within 42 days. * Must not have received monoclonal antibodies within at least 3 half-lives of the antibody after its last dose. * Must not have received radiotherapy (XRT) within 14 days prior to study drug administration (small field) or 42 days for craniospinal XRT, or if \>=50% radiation of pelvis. * At least 84 days must have elapsed after stem cell infusion prior to study drug administration * No evidence of active graft-versus-host disease (GVHD) and at least 100 days must have elapsed after allogeneic bone marrow transplant or stem cell infusion prior to study drug administration * Participants must have adequate bone marrow function, defined as: * Peripheral absolute neutrophil count (ANC) \>=1.0\*10\^9/liter (L). * Platelet count \>=100\*10\^9/L (not receiving platelet transfusions within a 7-day period prior to study drug administration). * Hemoglobin (Hb) at least 8.0 grams per deciliter (g/dL) at baseline (blood transfusions are allowed during the screening period to correct Hb values \<8.0 g/dL). * Participants must have adequate renal function, defined as: * A serum creatinine based on age/gender, derived from the Schwartz formula for estimating glomerular filtration rate (GFR), per protocol-specified criteria. * Serum creatinine clearance or GFR \>=50 milliliters/minute/1.73 m\^2, based on a 12 or 24 hours (h) urine creatinine collection. * Participants must have adequate liver function, defined as: * Bilirubin (sum of conjugated + unconjugated) \<=1.5 times the upper limit of normal (ULN) for age. * Alkaline phosphatase, alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \<=3\*ULN (in the case of liver metastases \<=5\*ULN), unless there are bone metastases, in which case liver-specific alkaline phosphatase must be separated from the total and used to assess the liver function instead of the total alkaline phosphatase. * Serum albumin \>=2 g/dL. * All participants and/or their parents or guardians must sign a written informed consent. * Participants must be willing to comply with all aspects of the protocol. Exclusion Criteria: * Females who are breastfeeding or pregnant at Screening or Baseline. A separate baseline assessment is required if a negative screening pregnancy test was obtained more than 72 h before the first dose of study drug. * Females of childbearing potential who: * Do not agree to use a highly effective method of contraception for the entire study period and for 6 months after study drug discontinuation, that is: * Total abstinence (if it is their preferred and usual lifestyle) * An intrauterine device (IUD) or intrauterine system (IUS) * A contraceptive implant * An oral contraceptive OR * Do not have a vasectomized partner with confirmed azoospermia. * Males who have not had a successful vasectomy (confirmed azoospermia) or they and their female partners do not meet the criteria above (that is, not of childbearing potential or practicing highly effective contraception throughout the study period or for 3 months after study drug discontinuation). No sperm donation is allowed during the study period or for 3 months after study drug discontinuation. * Concomitant Medications: * Participants receiving corticosteroids who have not been on a stable dose for at least 7 days prior to study drug administration. * Participants who are currently receiving other anticancer agents. * Participants who are receiving cyclosporine, tacrolimus or other agents to prevent GVHD post bone marrow transplant. * Participants who are receiving strong cytochrome P450 3A4 (CYP3A4) inhibitors and inducers including traditional herbal medicinal products (example, St. John's Wort). * Phase 1: Received prior therapy with eribulin mesilate within 6 months prior to study drug administration. * Phase 2: Received prior therapies with eribulin mesilate or irinotecan hydrochloride (for prior irinotecan hydrochloride, participants can be included if there was no tumor progression during irinotecan therapy). * Any other malignancy that required treatment (except non-melanoma skin cancer, or histologically confirmed complete excision of carcinoma in situ), within 2 years prior to study drug administration. * Has hypersensitivity to either study drug or any of the excipients. * Has a known prior history of viral hepatitis (B or C) as demonstrated by positive serology (presence of antigens) or have an uncontrolled infection requiring treatment * Has \>Grade 1 peripheral sensory neuropathy or \>Grade 1 peripheral motor neuropathy graded according to the Modified ("Balis") Pediatric Scale of Peripheral Neuropathies. * Has cardiac pathology, defined as: * Participants with known congestive heart failure, symptomatic or Left ventricular (LV) ejection fraction \<50% or shortening fraction \<27% and participants with congenital long QT syndrome, bradyarrhythmias, or QT interval (QTc)\>480 milliseconds on at least 2 separate electrocardiograms (ECGs). * Has CNS disease: Participants with brain or subdural metastases are not eligible unless the metastases are asymptomatic and do not require treatment or have been adequately treated by local therapy and have discontinued the use of corticosteroids for this indication for at least 28 days prior to study drug administration. Participants must be clinically stable. It is not the intention of this protocol to treat participants with active brain metastases. Note: Screening CNS imaging for participants with a known history of CNS disease is required. * Have had or are planning to have the following invasive procedures: * Major surgical procedure or significant traumatic injury within 28 days prior to study drug administration. * Laparoscopic procedure or open biopsy within 7 days prior to study drug administration * Central line placement or subcutaneous port placement is not considered major surgery. * Core biopsy, including bone marrow biopsy within 2 days prior to study drug administration. * Fine needle aspirate within 3 days prior to study drug administration. * Participants with known human immunodeficiency virus (HIV); due to lack of available safety data for eribulin therapy in HIV infected participants. * Has any serious concomitant illness that in the opinion of the investigator(s) could affect the participant's safety or interfere with the study assessments (including active or severe chronic inflammatory bowel disease or bowel obstruction). * Has received a live-virus vaccination within 30 days of planned start of study therapy. Seasonal flu vaccines that do not contain live virus are permitted.