How do I periodically run a command with very short interval and get the return?

Why do you think 'watch' will not work?

$ cat periodic.sh
#!/bin/bash
echo $(date)
$ chmod +x periodic.sh

$ watch -n 5 ./periodic.sh

(
  while true
  do
    your-command-here
    sleep 5
  done
) &
disown

Maybe with nohup? It's designed to let a job run after the shell is closed.

You can also use screens.

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