Something like an Ibeam could, it would just require something closer in size/larger than a helicopter. A 50 foot Ibeam is a bit over 2 tons; a small modern military helicopter would be able to carry that easily. When I say drone I just mean a remotely controlled copter-thing, not specifically a small one.

An entire building is less feasible unless it is small, but it could be done using modular components

If building costs go way up, so do rents for normal people (and less gets built). Just something to consider.

For building construction that is substantial, a 50 foot wide flange section is probably going to be 1.5 to 4 times heavier than your 4000 lbs, in my experience.

Are we discussing this as a hypothetical, or as a practical building method?

Plus I don't really want my town to sound like the set of Apocalypse Now.

Fleets of large and noisy helicopters hauling cargo over a city all day sounds way worse than regular road traffic...

It's a waste of money but it is not at all "100%" out of the question. If you're willing to pay for the extravagance of air travel, you can make it happen. There are many many people and organizations capable of paying it.