Archimedes referenced a technology that was absolutely not possible for his time and civilization. The entire planet working together with every resource at their disposal could not have enacted his proposal. That, to me, says "technical" problem. Our civilization COULD get to Mars. We could do it for far less than the GDP of a single nation. Even if that's not a rational or sensible use of our resources, we could do it, without impacting the survival of our species whatsoever. That, to me, makes it a social / financial issue rather than a technical one. I agree with most of your points, though (even if I'm going to argue with some of them below, just because ) so I don't think we have vastly differing opinions. We just have different terminology is all.

Fuel is, of course, a gigantic issue. And for what it's worth, I'm on your "not worth it" side. I do think you're vastly exaggerating the problems, however. Why on Earth Mars do we need a BILLION kg? People have lived on the ISS for upwards of a year, and the entire ISS is only about half a million kg. Or, in other words, two thousand times smaller. You seem to be basing most of your mass requirement on shielding the entire capsule, which is not necessary. Radiation shelters work adequately on the ISS, and would likely do the same on a Mars trip. Once the colonists land, they can use Martian soil and the atmosphere to provide additional shielding. Also, as to launch costs, my understanding is that LEO costs of the Falcon Heavy are around $2000/kg, not $10,000.

So, yeah. Stupid and unnecessary? Probably. 10 trillion dollars for a reasonable shot at success? No way. A megaton is equal to about five of the world's largest cruise ships. That is an absolutely absurd amount of mass for the bare-bones existence of four individuals.