CIDADE ESTRUTURAL, Brazil—Sandra Montes, 32, once dreamed of being a famous singer, but her hopes were crushed by a recession that Brazil is still struggling to overcome. So she turned to rummaging through Latin America’s largest dump looking for recyclables to sell, clothes to wear and toys to give her children.

Now she and her fellow scavengers at the Estrutural landfill will have to find work elsewhere. Nearly 60 years after Brazil carved a modernist capital out of its wind-swept central plateau and began dumping its solid...