I've read about this neat process for extracting CO2 from the atmosphere. You place a ultra compact box in the ground containing it's own building instructions and tooling for separating CO2 in to oxygen and building giant structures of the carbon. It's apparently called photosynthesis.

You're joking, but producing biochar is actually a decent way of doing carbon capture that doesn't require any fancy technology. You even get some energy out of it. Unfortunately you'd need to plant absolutely massive amounts of trees to put a dent into the trillion tonnes of CO2 we've released in the last 150 years.

Also algae could be neat allies. But there we are playing with fire, maybe GMO algae can be engineered to be supper efficient, but at the risk of compromising marine ecosystems.

The fern Azolla already did this once! linky --> https://en.wikipedia.org/wiki/Azolla_event

There's a lower-tech way to get algae on board - feed them the micronutrients they're limited by in the ocean, which is mostly iron:

https://en.wikipedia.org/wiki/Ocean_fertilization

https://en.wikipedia.org/wiki/Iron_fertilization

It is far cheaper to reduce the emissions of CO2 by switching to carbon free energy to a sufficient level that atmospheric CO2 stops rising (~25-50% of current level, different sources give different numbers), than to extract CO2 from the atmosphere.

You are now talking about the cost of the technology, but that's not the complete picture. Switching to carbon free energy will probably work, but it will take longer to get to where we want to go. If it takes longer to stop climate change, there will likely be higher costs in undoing the effects of it. There are already severe droughts and highly unusual weather right now, if it takes us another 20 years to undo what we've done, who knows what kind of extreme weather we will see?

However, if we can prevent (more years of) extreme weather and rising sea levels by extracting CO2 from the atmosphere, then the total bill for the whole project will be lower because we don't have to pay for undoing the damage.

Additionally, who says that money spent on climate change prevention is money wasted? I'm fairly convinced that these investments will actually create jobs, in addition to preventing disaster.

Bill mentions it at the 15 minute mark [1]. He says he's the biggest funder of various carbon capture technologies, liquids, solids, you name it. He then explains that it's 2 orders of magnitude non-economic and will have to be 10x the size of the current fossil industry to succeed at the rate people are saying is needed.

[1] https://www.youtube.com/watch?v=d1EB1zsxW0k&t=900s

MIT Technology Review published a piece on one possible CO2 extraction solution just a few days ago. Besides being somewhat expensive, it also needs to occupy substantial physical area.

https://www.technologyreview.com/s/612928/one-mans-two-decad...