C14 Study in Oncology Patients With Advanced and/or Metastatic Solid Tumors

Inclusion Criteria: * Have a histological or cytological diagnosis of cancer (solid tumor), with clinical or radiologic evidence of locally advanced and/or metastatic disease, for which no life-prolonging therapy exists (that is, refractory to standard therapy and/or therapies known to provide clinical benefit, or for which no standard therapy exists). Note: participants who have had progressive disease after receiving pemetrexed for metastatic disease are excluded from receiving the combination with pemetrexed during the safety extension study. Participants who have had progressive disease after receiving gemcitabine for metastatic disease are excluded from receiving the combination with gemcitabine during the safety extension study. * Have a body surface area greater than or equal to 1.37 meters squared (m\^2) * Have given written informed consent prior to any study-specific procedures * Adequate hematologic, hepatic and renal function * Have a performance status of less than or equal to 2 on the Eastern Cooperative Oncology Group (ECOG) scale * Have discontinued all previous treatments for cancer, including chemotherapy, radiotherapy, anticancer hormone therapy, or other investigational therapy for at least 30 days prior to study entry and recovered from the acute effects of therapy (at least 42 days for mitomycin-C or nitrosoureas, or 60 days for monoclonal antibodies) * Are reliable and willing to make themselves available for the duration of the study and are willing to follow study procedure * Males and females with reproductive potential: Must agree to use medically approved contraceptive precautions during the study and following the last dose of study drug until, in the judgment of the investigator, it is safe for the participant to become pregnant or father a child * Females with childbearing potential: Have had a negative serum pregnancy test less than or equal to 7 days before the first dose of study drug and must also not be breastfeeding * Have an estimated life expectancy that, in the judgment of the investigator, will permit the participant to complete 1 full cycle of treatment (beyond the initial \[\^14C\]LY2603618 dose) * Prior radiation therapy for treatment of cancer other than pancreatic is allowed to \<25% of the bone marrow and participants must have recovered from the acute toxic effects of their treatment prior to study enrollment. Prior radiation to the whole pelvis is not allowed. Prior radiotherapy must be completed at least 4 weeks before study entry. Exclusion Criteria: * Have received treatment within 28 days of the initial dose of study drug with an experimental agent for noncancer indications that has not received regulatory approval for any indication * Have previously completed or withdrawn from this study or any other study investigating LY2603618 or any other checkpoint kinase one (Chk1) inhibitor * Have a known allergy to gemcitabine, pemetrexed, LY2603618, or any ingredient of gemcitabine, pemetrexed, or LY2603618 (like Captisol) * Have serious preexisting medical conditions (left to the discretion of the investigator) other than advanced cancer * Have symptomatic central nervous system (CNS) malignancy or metastasis (screening not required). Participants with treated CNS metastases are eligible for this study if they are not currently receiving corticosteroids and/or anticonvulsants, and their disease is asymptomatic and radiographically stable for at least 90 days * Have current hematologic malignancies or either acute or chronic leukemia * Have an active fungal, bacterial, and/or known viral infection including human immunodeficiency virus (HIV) or viral (A, B, or C) hepatitis (screening is not required) * Have a QTc interval of \>500 milliseconds (msec) on the screening electrocardiogram (ECG) * Have ECG abnormalities on the screening ECG such as significant conduction abnormalities, ischemic changes (such as prior Q-wave myocardial infarction and/or marked ischemic ST- and T-wave), arrhythmias (such as persistent or paroxysmal ventricular or supraventricular arrhythmias, including atrial fibrillation), or other ECG abnormalities that would put the participant at unnecessary risk in the opinion of the investigator * Have participated in a \^14C study within the last 6 months prior to screening for this study. The total exposure from this study and the previous study must be less than 5 milliSieverts (mSv).