Long-Term Unemployment Elimination Act of 2021 This bill creates a program to generate job opportunities for full-time students and certain workers who have been unemployed for at least 27 weeks. Specifically, the Department of Labor must establish and administer a jobs initiative for eligible workers (i.e., individuals who are at least 18 years of age, are authorized to be employed in the United States, have not been employed or are full-time students for at least 27 weeks, and are currently seeking or have been seeking employment for at least 4 weeks), carried out through local boards or community-based organizations. Labor shall award grants to local boards and community-based organizations to carry out jobs programs, including any added costs related to them in order to achieve economic development and job growth. Priority shall be given to local boards proposing to serve (1) high-poverty areas; or (2) areas for which the prime working-age employment-to-population ratio has been low, relative to that ratio for the United States for at least three years.