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No, it usually has 70% to 90% less CO2 emissions compared to fossil based fuel, because there are CO2 emitters in the supply chain: Transportation, heating, food for workers, etc. This is the same reason renewable energy has a small CO2 footprint as well. However for e-fuels you need about 6 times as much energy to go the same distance with an electric vehicle. So you have to build 6 as many renewables (only then do e-fuels make sense) compared to electric cars.
Are you just starting out? I got started with home labbing with a Raspberry Pi 2B (1GB RAM!) and an external HDD I had lying around. I host Yarr, Navidrome, backups and a dashboard app Ive written on there and I am quite satisfied. I would really recommend starting small with hardware you already have and then buy new hardware as you go along. I am also using Tailscale. With this you can get your initial setup up and running in a day and save money if it turns out home labbing isnt for you or you dont really need the hardware.