Which of the following is a task of the County Board of Elections?

The task of appointing district board workers falls under the responsibilities of the County Board of Elections. This board is crucial for overseeing the electoral process within the county, and one of its key functions is to ensure that there are adequate personnel available to manage polling places on Election Day, facilitate the voting process, and guarantee that elections run smoothly.

The other options are outside the scope of the County Board of Elections' functions. Determining voter eligibility, while related to the electoral process, is primarily the responsibility of the state's election officials and often involves various departments. Issuing marriage licenses is managed by the local registrar’s office or county clerk, not the election board. Organizing community events does not relate directly to election duties and is usually handled by local governments or community organizations. Thus, the specific role of appointing district board workers distinctly aligns with the responsibilities of the County Board of Elections.