The issues are these: How much energy does it take to capture the gas (CH4 or CO2) and what do you do with it then? Capture is not much of a challenge but what comes next is. Machines can capture but there really is no place to put the gasses so they need to be treated to render them harmless. CO2 + H2O = Carbohydrate is the best equation I can think of. This happens in green plants and the answer to the CO2 problem is photosynthesis. It’s not sexy, there’s no Manhattan Project or Space Race feel to it and no big machines to gush over, but it’s our best bet from perspectives of cost of energy and long term stability.
CH4 is coming out of the permafrost at alarming rates because it’s thawing and all the capture ideas we might have won’t change that. Preventing the thaw requires taking CO2 out of the air and I estimate we need to do it at the level of 1 trillion tons, a doable thing in 10 years if we double the amount of photosynthesis occurring on the planet.
Photosynthesis isn’t perfect, it will require recurring attention because we can only manage carbon this way, like a diabetic on insulin, but that’s what we’re stuck with. Carbon is a management problem not something to be defeated.
Methane produced by agriculture is not the problem, that’s a part of the short carbon cycle. Methane from permafrost comes from the long carbon cycle and it is that cycle that we need to attend to.