Study to Assess Safety and Preliminary Activity of Eribulin Mesylate in Pediatric Participants With Relapsed/Refractory Rhabdomyosarcoma (RMS), Non-rhabdomyosarcoma Soft Tissue Sarcoma (NRSTS) and Ewing Sarcoma (EWS)

Inclusion Criteria: * Age: ≥12 months to \<18 years old at the time of informed consent * Diagnosis: Histologically confirmed rhabdomyosarcoma (RMS), non-rhabdomyosarcoma soft tissue sarcoma (NRSTS) (Grade 2 or 3), or Ewing sarcoma (EWS) which is relapsed or refractory (failed front line therapy) * The presence of measurable disease meeting the following criteria: * At least 1 lesion of ≥1.0 centimeter (cm) in the longest diameter for a non-lymph node or ≥1.5 cm in the short-axis diameter for a lymph node that is serially measurable according to Response Evaluation Criteria In Solid Tumors (RECIST) 1.1 using computerized tomography/magnetic resonance imaging (CT/MRI). * Lesions that have had radiotherapy must show subsequent radiographic evidence of increase in size by at least 20% to be deemed a target lesion. * Therapeutic options: Participant's current disease state must be one for which there is no known curative therapy or therapy proven to prolong survival with an acceptable quality of life. * Performance level: Performance score ≥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 performance score. * Participants must have fully recovered from the acute toxic effects of all prior anticancer therapy and must meet the following minimum duration from prior anticancer directed therapy prior to study drug administration. If, after the required time frame, the numerical eligibility criteria are met, eg, blood count criteria, the participant is considered to have recovered adequately: * Cytotoxic chemotherapy or other chemotherapy known to be myelosuppressive: ≥21 days after the last dose of cytotoxic or myelosuppressive chemotherapy (42 days if prior nitrosourea). * Anticancer agents not known to be myelosuppressive (eg, not associated with reduced platelet or absolute neutrophil count \[ANC\] counts): ≥7 days after the last dose of agent. * Monoclonal antibodies ≥ 3 half-lives must have elapsed from infusion of last dose of antibody (including checkpoint inhibitors), and toxicity related to prior antibody therapy must be recovered to Grade ≤1. * Hematopoietic growth factors: ≥14 days after the last dose of a long-acting growth factor (eg, Neulasta) or 7 days for a short-acting growth factor. For agents that have known adverse events (AEs) occurring beyond 7 days after administration, this period must be extended beyond the time during which AEs are known to occur. The duration of this interval must be discussed with the sponsor. * Interleukins, interferons, and cytokines (other than hematopoietic growth factors): ≥21 days after the completion of interleukins, interferons, or cytokines (other than hematopoietic growth factors) * Stem cell infusions (with or without total body irradiation \[TBI\]): ≥84 days * Allogeneic (non-autologous) bone marrow or stem cell transplant, or any stem cell infusion including donor lymphocyte infusion or boost infusion: ≥84 days after infusion and no evidence of graft versus host disease (GVHD) * Autologous stem cell infusion including boost infusion: ≥42 days * Cellular therapy: ≥42 days after the completion of any type of cellular therapy (eg, modified T-cells, natural killer cells, dendritic cells, etc) * Radiation therapy (XRT)/External Beam Irradiation including Protons: ≥14 days after local XRT; ≥150 days after TBI, craniospinal XRT or if radiation to ≥50% of the pelvis; ≥42 days if other substantial BM radiation * Radiopharmaceutical therapy (eg, radiolabeled antibody, 131I-metaiodobenzylguanidine): ≥42 days after systemically administered radiopharmaceutical therapy. * Adequate bone marrow function, defined as: * ANC ≥1.0 × 10\^9/Liter (L) * Platelet count ≥100 × 10\^9/L (transfusion independent, defined as not receiving platelet transfusions within a 7-day period prior to study drug administration) * Hemoglobin at least 8.0 grams per deciliter (g/dL) at Baseline (blood transfusions are allowed during the screening period to correct hemoglobin values less than 8.0 g/dL) Note: As blood transfusions are permitted to meet the hemoglobin criteria, participants requiring transfusion must not be known to be refractory to red blood cell or platelet transfusions. * Adequate renal function, defined as: * A serum creatinine based on age/gender, derived from the Schwartz formula for estimating glomerular filtration rate (GFR) * Or creatinine clearance or GFR ≥50 milliliters per minute (mL/min)/1.73 meters squared (m\^2) based on a 12 or 24 hour urine creatinine collection * Adequate liver function, defined as: * Bilirubin (sum of conjugated + unconjugated) ≤1.5 × upper limit of normal (ULN) for age * Alanine aminotransferase (ALT) ≤110 units per Liter (U/L). For the purpose of this study, the ULN for ALT is 45 U/L * Serum albumin ≥2 g/dL * Informed consent: All participants and/or their parents or legally authorized representatives must sign a written informed consent. Assent, when appropriate, will be obtained according to institutional guidelines. Participants must be willing to comply with all aspects of the protocol. Exclusion Criteria: * Pregnancy, breastfeeding, contraception: Females who are breastfeeding or pregnant at Screening or Baseline (as documented by a positive beta-human chorionic \[β-hCG\] or human chorionic gonadotropin \[hCG\] test with a minimum sensitivity of 25 International Units per Liter \[IU/L\] or equivalent units of β-hCG \[or hCG\]). A separate baseline assessment is required if a negative screening pregnancy test was obtained more than 72 hours before the first dose of study drug. * Females of childbearing potential (all post pubertal females will be considered to be of childbearing potential unless they have early menopause \[amenorrheic for at least 12 consecutive months, in the appropriate age group, and without other known or suspected cause\] or have been sterilized surgically \[ie, bilateral tubal ligation, total hysterectomy, or bilateral oophorectomy, all with surgery at least 1 month before dosing\]) 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, ie: * Total abstinence (if it is their preferred and usual lifestyle); * An intrauterine device (IUD) or intrauterine system (IUS); * A contraceptive implant; * An oral contraceptive (must be on a stable dose of the same oral hormonal contraceptive product for at least 4 weeks before dosing with study drug and for the duration of the study and for 6 months after study drug discontinuation); or * Do not have a vasectomized partner with confirmed azoospermia. For sites outside of the European Union (EU), it is permissible that if a highly effective method of contraception is not appropriate or acceptable to the participant, or the participant has commenced/adjusted/changed oral hormonal contraceptive product/dose within 4 weeks prior to study drug administration, then the participant must agree to use a medically acceptable method of contraception, ie, double barrier methods of contraception such as condoms plus diaphragm or cervical/vault cap with spermicide. * Males who have not had a successful vasectomy (confirmed azoospermia) or if they and their female partners do not meet the criteria above (ie, 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: * Corticosteroids: Participants receiving corticosteroids who have not been on a stable or decreasing dose of corticosteroid for at least 7 days prior to study drug administration (except when indicated for Central Nervous System \[CNS\] metastases, then participants must not have received corticosteroids for at least 28 days) * Anticancer Agents: participants who are currently receiving other anticancer agents * Anti-GVHD agents Post-transplant: Participants who are receiving cyclosporine, tacrolimus or other agents to prevent graft-versus-host disease post bone marrow transplant * Strong CYP3A4 inducers/inhibitors * Received prior therapy with eribulin mesylate * Any other malignancy that required treatment (except for non-melanoma skin cancer, or histologically confirmed complete excision of carcinoma in situ), within 2 years prior to study drug administration * Has hypersensitivity to eribulin or any of the excipients * Has a prior history of viral hepatitis (B or C) as demonstrated by positive serology (presence of antigens) or have an uncontrolled infection requiring treatment. Participants with a known prior history of hepatitis B or C may be eligible pending agreement with the sponsor. * 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: Participants with known congestive heart failure, symptomatic or left ventricular (LV) ejection fraction \<50% or shortening fraction \<27% * Participants with congenital long QT syndrome, bradyarrhythmias, or QTc \>480 millisecond (msec) 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 (eg, surgery or radiotherapy) and have discontinued the use of corticosteroids for this indication for at least 4 weeks prior to study drug administration. Confirmation of radiographic stability must be done by comparing the brain scan (CT or MRI) performed during the Screening Period, using the same imaging modality, to a brain scan performed earlier (and following local therapy where applicable). Participants must be clinically stable. It is not the intention of this protocol to treat participants with active brain metastases. Note: CNS imaging is required to confirm eligibility for participants with a known history of CNS disease. * 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 but must be placed at least 2 days prior to study drug administration * 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 * 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 * Participants with known human immunodeficiency virus (HIV); due to lack of available safety data for eribulin therapy in HIV-infected participants