The meltdown left a simmering stew of toxic radioactivity under the rubble, covered by a hastily built shelter that's crumbling.
"There's still a massive inventory of radionucleides inside the shelter - and the shelter is far from being airtight," said project manager Laurin Dodd.
Work is finally underway on a permanent solution, but Chernobyl today is still a very dangerous place.
Special protective clothing is required. The radiation level is so high that you can't stay long.
The construction equipment cabs have lead sheeting; every bucket of rubble is monitored for radiation.
The solution, 10 years in the planning, is an enormous steel arch, to be built in sections, then moved on tracks over the reactor.
At 345 feet, it'll be taller than the statue of liberty - and wider, at 840 feet, than the St. Louis Gateway Arch.
Not only is the project huge, but so is the cost: almost $1.5 billion. And the United States is the largest-single country donor. Why? Not just to help Ukraine, but also to help guarantee the future of nuclear power.
"Nuclear power will always have a shadow over it as long as Chernobyl is a message of concern," said U.S. ambassador to Ukraine William B. Taylor.
CBS News was on the site less than 10 minutes when one member of the group went over his exposure limit.
"Right now the dose rate is 200 times the background of what you'd have in Washington, D.C.," Dodd said.
The steel arch is supposed to keep the radiation contained for at least 100 years - while future generations figure out how to dispose of the mess.