Not impossible!
Hugely expensive, yes. But so was Apollo, and what did we get out of that? A few kilos of rock.
For the space elevator, you don't use *one* cable. Not unless you're suicidal, anyway.
For terraforming Mars (the payoff from which is an entire planet to colonise - with more total space than all of North and South America combined, and Europe seems to have managed that nicely, even with the active opposition of the natives).
To create the atmosphere, we bombard Mars with icey material from Saturn's rings - expensive to get started, and a long-term (probably a century at least) project, but it *can* be done. This also provides some heat input (friction heating of the incoming "ice bombs") and more water. The water is likely enough to put paid to the sandstorms. If it's not, then plant beach grass.
To protect the atmosphere, and any future inhabitants, from Solar radiation, a Mylar balloon is constructed at roughly 15km altitude, enclosing the entire planet, with a few strategically-placed openings for space traffic.
And there's more than enough sunlight to grow crops. We do it just fine in high latitudes on Earth.
Expensive? Sure. Payoff? An entire *planet*. How much is that worth?