meybe You can run https://www.fuzix.org/ on this? how long this device work in real situations? a week, month?

This is a terminal for accessing another device. I use it with ESP-01s, and I think it might be possible to get Fuzix working on an ESP-01, but you can use any device with a serial port.

I get up to 18 hours (less if I'm using wifi) out of good-quality AA batteries.

In my opinion, the key issue is electricity. Fuzix will work on very weak equipment. But no one has made a mobile computer yet.

I am writing this as a person who is looking for such a small computer that would work for a week or a month. does not need much beyond a normal system.