Is there any reason no one is talking about terraforming seriously yet? What kind of technologies can be used to draw the CO2 out of the atmosphere, how much would it cost to make a difference?
A few people are talking about "geoengineering" approaches, e.g. pumping sulphur dioxide into the upper atmosphere to cause a cooling effect. But such ideas are pretty fringe, I suspect because a) they sound crazy b) that kind of intervention risks serious unexpected consequences.
Basic thermodynamics can provide a lower limit : at least 9x more energy than was gained by burning it (except for oil burned in power plants, where it'd be 6-7x, but that's not much).
That seems to mean that without massive expansions in nuclear power it's just not in the cards.
But like with nuclear power, we don't have to do all the work. We don't need to generate electricity and then run conventional a CO2 filter. Hypothetically, if we dump 1 trillion dollars into genetic engineering, why can't we design some biological chain reaction that would do all the work?
Actually I sort of like the medieval approach : government prizes. We want this done, why not create a competition ? If a company can solve this problem, document how you've done it (AFTER the fact) and present it to a board of government scientists. If they believe you, you get the prize money.
And make the prize money something like $5 billion or so. Chump change for what you're asking for. Besides, it has to be an amount that would provide decades of comfort for a company of at least a dozen people + a nice reward for any investors.
This worked for draining swamps in the late middle ages. It worked for building things like clocks and compasses, and these sorts of prizes is how ocean-capable ships were designed in the first place. I would argue that the (early) prizes were for 'allow a ship to navigate without any visible markers on the ground', about as nebulous a concept as 'lower atmospheric co2 by 50% - or at least prove you can do it' would be.