Is there a way to build a closed loop system that would suck CO2 and make methane from it for example? Methane can be used for further energy production but again with closed loop, capturing all produced CO2.
That part's easy. Where's your energy source?
The whole issue is not Co2, it's energy.
Solar + methane itself fed back into the loop.
This means you are going to need to pump more energy in than you produce. Which begs the question GP asked.
Put another way, perpetual motion machines are impossible. What you propose is that.
Put some nice nuclear batteries on this and you might have a chance. But net power in is greater than net power out, again due to thermodynamics. So ...
Efficiency is a problem though. With all the steps involved, the laws of thermodynamics become very punishing.