How to avoid having shell scripts which fail from killing Emacs shell?

[David Karr]

I've often been annoyed by shell scripts which fail for particular
reasons, at which point it causes my Emacs shell buffer to get killed,
with "Process shell<2> finished".  I think it's possible I could code
the script to make this not happen, but it would make more sense
(especially when I didn't write the script) to configure Emacs or the
Bash inside Emacs, so that it doesn't get killed when a script it's
running fails.

What are my options for mitigating this?