If radium has such a long half-life, how can radon possibly be a threat to us?
There are two factors here:


Radon is a gas and can thus enter the body very easily, through inhalation.
$\ce{^222Rn}$ and its first 4 decay products are extremely radioactive, with halflives of miliseconds to a few days, releasing alpha and beta radiation.


When the two are combined, we have a problem. Even if the quantity is low, inhaling radon places an extremely radioactive substance at very close proximity to the cells in one's lungs. Normally, alpha and beta radiation are relatively harmless as they cannot penetrate tissue very well, but now they have direct access. All it takes is one mutation in the wrong place to cause lung cancer.

To give you an idea of what a difference these make, the volumetric activity which is one guideline for needing to mitigate radon accumulation in a house is around 800 $\mathrm{Bq/m^3}$, which corresponds to only 141 fg of radon per cubic metre of air. This isn't like a uranium mine or something, but at this level, a certain increased risk of cancer is expected. Apparently some study found a 50% increased risk at a much lower level ~150 $\mathrm{Bq/m^3}$, too.



N.B. a becquerel (Bq) is a reciprocal second—1 Bq means one atom of the radioactive substance disintegrates every second.