OLKILUOTO, Finland—In billion-year-old bedrock, 100 stories below ground, the Finns have found what the whole world is looking for: A place to bury their most dangerous nuclear waste.

From the U.S. to the U.K., Germany and Japan, plans to dig tombs for the highly radioactive fuel rods have hit political roadblocks or faced backlash from locals who don’t want toxic waste in their backyards.

But...