This is partially based on: Half Life Foundations of Chemistry and might show what the problem is, apologies if you know some or all of it already:
The half-life of any element is the time taken for half of it to decay to another lower number element (or elements).
For example, if one begins with a gram of carbon-10, 20 seconds later only half a gram will remain, after 40 seconds only a quarter gram will be left, after 60 seconds an eighth of a gram, after 80 seconds one sixteenth of a gram, and after 100 seconds have elapsed from the beginning of the experiment, only one thirty-second of the original carbon-10 will remain.
The Chernobyl problem stems from 3 (possibly 4) differences to the above decay pattern.
As you say, instead of 1g, we have 180 metric tons of uranium dioxide fuel.
The half life of the fuel is much longer than carbon, one component of the fuel has a half life of 4.5 billion years; that is, half the atoms in any sample will decay in that amount of time.
As it decays, it releases gamma rays, which have more than enough energy to break the DNA strands in human body cells, and produce both long term and short term serious health "issues".
Uranium-238 decays by alpha emission (a helium nucleus) into thorium-234, which itself decays by beta emission (where a neutron is converted into a proton, an electron, and an electron antineutrino) to protactinium-234, which decays by beta emission to uranium-234.
So there is still radioactivity from the by-products.
The source for this picture: Half Life Diagram provides more information on the decay process:
Black humor though it might be, considering the immense bravery of the people involved in the cleanup, the story/urban myth goes that they have road signs near the plant that say :
Chernobyl Region: Close your windows and drive very fast.