TL;DR
I booted Debian Linux on a 4-bit intel microprocessor from 1971 - the first microprocessor in the world - the 4004. It is not fast, but it is a real Linux kernel with a Debian rootfs on a real board whose only CPU is a real intel 4004 from the 1970s. The video is sped up at variable rates to demonstrate this without boring you. The clock and calendar in the video are accurate. A constant-rate video is linked below.
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But does it run Doom?
If it is “full Linux” then the answer is clearly yes. Not very fast I suppose.