To avoid any concerns about scalability, as well as about energy supply intermittency, I made my estimate using only the oldest process in the chemical industry, lime burning, which predates plastics, oil drilling, steel, iron, bronze, writing, cities, ceramic, and possibly even agriculture. The only difference from the Neolithic method is that you have to retort the lime in a sealed chamber so you can collect the carbon dioxide it absorbed from the atmosphere after the last time you calcined it. This doesn't require an enormous amount of machinery, just very large machinery. Basically, a giant tin can similar to a water tower, maintained at a mildly negative gauge pressure as it's heated up.
My estimate, IIRC, was that, without soda for process intensification, you would need an amount of limestone a few times larger than the amount mined by the cement industry every year. That's fine, though; limestone is about 20% of all sedimentary rock, and sedimentary rock is 73% of Earth's land surface and 8% of the entire crust.
Very roughly, Earth is 6e24 kg, her crust is 6e22 kg, and its limestone is 1e21 kg. By contrast, the 427 ppm of carbon dioxide we need to capture half of is only 3.34 teratonnes, 3e15 kg. Limestone is 44% carbon dioxide by mass, so it absorbs 44% of its mass in carbon dioxide each time through the cycle. This absorption takes about 5 years if it's sitting in a paper bag dry, about a month when you whitewash a wall with it, or a second or so when you catalyze the absorption with a few percent of lye, like in a scuba diving rebreather.
The USGS publishes a lot of information about the world lime market at https://www.usgs.gov/centers/national-minerals-information-c.... From it, we can see that current US lime prices are 20¢/kg, 3.2 billion dollars for a total yearly production of 16 million tonnes. (They say that prices at the plant were as low as US$131/tonne in 02020, so probably the production cost is closer to 13¢/kg, including the cost of mining.) That's probably tonnes of CaO, which is the other 56% of limestone that isn't carbon dioxide. If burning lime in a giant tin can to capture the gas were about as expensive as how it's done today, that's about 24¢ per kg of carbon dioxide, not counting the cost of warehousing the resulting lime until it's absorbed the gas so you can calcine it again. (Remember, you can reuse the same lime every few hours if you dope it with a soda catalyst.)
World lime production is closer to 420 million tonnes per year, mostly of course in China, 26 times US production.
1.8 trillion tonnes of lime is 4300 years of total world production (or 120,000 years of current US production), so we're talking about a significant scale up. It's not just a few times global cement energy production. But it's still only two millionths of the limestone in the crust of the earth, and maybe you'd want to reuse the same lime many times so you don't have to mine it again.
But probably some process involving more sophisticated sorbents like triethanolamine, combined with point source capture, will end up being cheaper in the end. And see my notes in https://news.ycombinator.com/item?id=44461843 about accelerated olivine weathering. The lime approach serves only as an easily computable upper bound on difficulty.
The USGS mineral commodities summaries also have an entry for "stone (crushed)", which is 70% limestone. This is 1.5 billion tonnes per year in the US, which I guess is a billion tonnes of limestone, so 1.8 trillion tonnes of quicklime (3.4 billion tonnes of limestone) is only about 3400 years of US mine production. It only costs about 1–2¢/kg, which is in accordance with what I've seen. No world production figures are given, but we can probably guess that that's another thing China produces 20 or 30 times more of.
So, the cost of quicklime is almost all (>80%) calcination, and I believe that almost all of that number is energy. We can put an upper bound on its energy consumption based on that 13-cent cost in 02020: coal is often the cheapest source of the heat needed for calcination, and in 02020, it reached a low around US$60/tonne ( https://fred.stlouisfed.org/series/PCOALAUUSDA). So we know that producing a kg of quicklime can't require more than about 2 kg of coal, which provides 33MJ/kg or less (0.7¢/kWh or US$1.80/GJ), so 70MJ per kg of quicklime or of carbon dioxide. That's probably not a very tight upper bound, but I doubt it's high by more than a factor of 3.
Removing 1.4 teratonnes of carbon dioxide over 40 years is 1.1 million kg per second. At 70MJ/kg this multiplies out to 78 terawatts, roughly four times world marketed energy consumption.
Today, devoting a couple of terawatts to the problem would be unreasonably expensive, and tens of terawatts would require expanding world energy production considerably. At 24¢ per kg of carbon dioxide, removing 1.6 trillion tonnes of it would cost 400 trillion dollars, four years of world GDP (say, 10% of world GDP over 40 years). But that's just because the rollout of photovoltaic energy has just begun; the majority of the 18 terawatts or so of world marketed energy consumption is still supplied by fossil fuels, although they are clearly no longer cost-competitive with PV. PV manufacturing is still scaling up, though, and presumably PV energy production will exceed current world marketed energy consumption in a few years, and then continue to increase as new uses are found for the newly much cheaper energy.
78 terawatts is 0.04% of the 174 petawatts of terrestrial insolation.