Kevlar was introduced about the time Arthur Clarke wrote "Fountains of Paradise", with a popular science fiction plan for the elevator. It was about an order of magnitude away from the strength/weight necessary.
Today nanotube technology is approaching the point of necessary properties, but it will be decades before the massive production scale and uniform quality make the elevator possible.
The project is ten times the size of Apollo or Manhattan or the space station, so the biggest questions will be about economics and strength of leadership.
"...so the biggest questions will be about economics and strength of leadership."
The biggest question with any megaproject to advance Humanity is ALWAYS "strength of leadership".
As George-the-1st put it "...the Vision Thing..."
Though for the last 3 decades neo-Luddite neo-Malthusian antihumans love to pretend its all about [UNmoney] economics as their excuse for 'limits to growth' stasis-quo.
Expect immediate cries from the antihumans of "Even if we CAN do it it will 'cost' too much TO do it!!!"