Oh well...
I'm a Linux geek, but I understand some things.

Linux needs about 1MB memory to run in the very minimalistic shape. 256 BYTES is just not enough. External memory is slow and hard to use.

Linux is rather slow on 50MHZ hosts. 1MHZ makes it practically useless. If you plan on using the "external memory", divide the speed by factor of about 5.

Despite the fact there are patches to Linux that allow it to run on architectures without memory protection, it loses all the ability to survive program errors. In other words, expect kernel panic instead of core dump.

'51 C compilers are far from plain ANSI because of tricky architecture.

So, answering your question... Yes, it is possible.
It is possible to push a camel through the hole of a needle, but it requires a lot of work, resources and will change the camel a lot in the process.
Feel free to try.