EMBARRASS, Minn.—An earthen dam is set to rise behind the trees of Dan Ehman’s 120 woodland acres in northeastern Minnesota’s Iron Range, a region with close ties to mining for more than a century.

The planned dam, designed to hold back hundreds of millions of tons of mining waste, will be similar in structure and height—soaring 250 feet above Mr. Ehman’s century-old log cabin—to one in Brazil that burst in January, killing 270 in a tsunami of sludge.

That...