Grow woody plants, then convert them to carbon in a charcoal oven.

The most efficient way to capture CO2 is still to let trees/plants (some studies say that some kinds of grass can fix Carbon faster than trees) grow and then by high temperature (without oxygen) or radiation or whatever, sterilize it and landfill then in an isolated landfill. Basically recreating the process of mineral carbon production. I think this is the only feasible way to do it.
Limestone is another way that nature does this ( piling up shells) to do it, but I don't know any feasible process.
Filtering directly from atmosphere is not feasible because in gas form it would not be easy to contain, plus the low concentration makes it a p. in the ass to extract it.

I read some studies that producing iron/steel using ONLY vegetal coal/biofuel/biogas as fuel in all could be an industrial way to affectively capture carbon, as Iron and steel will keep some of the carbon in the metal and the biofuel used is only circulating carbon.

Problem is that unless you are landfilling trees in deep and very compact soil, all the ways to capture carbon are finite, a forest area will capture carbon until it is completely developed, after that it becomes neutral.
even building houses with wood is not an option as you will eventually need to repair, change and discard that wood that will be burned or landfilled in a place that will let the decomposition carbon escape.

Well here is another idea. Deep landfilling discarded wood.

But as you see, no solution is economic or even cheap.

And even for this deep landfilling if you use fossil fuel or fossil powered electric grid for your machinery, it will probably emit more than capture.

Ok thanks for all the comments. So here is incentive $175k/pound for high grade carbon nanotubes. Carbon fuel could pay for the system. Lots of carbon graphemes on google new inventions every day lately. Carbon based fertilizer. I just want to erase my carbon footprint assuming no one else will.

Last I heard such devices were large, expensive and inefficient. Otherwise, they would be employed everywhere already.

Erm nuclear reactor power plant for electricity?

As someone who's college had a running nuke plant I am thinking this is tongue in cheek.

That or you have a team of nuke engineers on hand?

I think the Skywalkers has that down with their moisture farming. well, until the Storm Troopers killed them off.

JK.
I hate to comment on something I've only recently started to study. however, And forgive me for not remembering the names, or published papers at this time.There has been the study of how photosynthesis actually works for quite some time. To farther, how to reproduce that effect in a lab, or by mechanical means. That said, we are very much just scratching the boundries of that.
I only bring that up because, if figured out. That potential would dwarf pretty much every other 'alt' energy.

Oh, and why I mentioned this, was because of the carbon footprint being laid. A photosynthesis based capture of energy would have to entail plants, or at least something similar, and we all know how precious that is.
Sorry, just thinking out loud.