* Render extinct-in-the-wild (or just plain extinct) a great many species of plants, fungi, and even a few animals, many of which have a rather large range and are difficult to kill.
* For whatever you miss in step one either because the precursors have legitimate uses we can't do without, or because we find we just can't kill them all, you need to tightly regulate and enforce a strict prohibition on those precursors in order to track who is doing what with what.
* Implementing the above step in a way that will really effectively reduce drug availability to nearly zero would require nothing short of global, endless warfare and a dystopian police state that would make Oceania look like - I don't know, Iceland? This is because some drugs are synthesized with very common materials and as such the only way to be sure no one is making them is to know what everyone is doing, all the time.
* To the extent you can't implement the previous step in full, you need to be sure the punishments meted out for failure to comply are especially harsh, so that enough people who think they can skate by under the radar are deterred.
* If you can do all this, you will drive the prices of drugs so high that the reward for manufacturing even a small quantity of something, provided you have the means to move the product, will definitely motivate people to try anyway. And if they achieve some success, they now have resources to fight you/bribe your allies.
So, doing the War on Drugs 'right', means giving up anything more than a pretense of freedom and democracy. Meanwhile, the cost of just legalizing the shit and treating addiction as a medical issue seems far less dire in comparison. When people say 'impossible' they mean 'impossible to do so in a way consistent with our values'.
Can you say the same for fighting fraud, speeding, and the sale of body organs? I think that you cannot.