How does someone exit from a sub routine without exitting altogether?

Using return.

sub doit {
   ...
   if (nothing_else_to_do) {
      return;
   }
   ...
}
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If you want to exit with a catchable error, you can use die or croak.

sub doit {
   ...
   if (did_not_provide_gizmo) {
      die("Gizmo required");
   }
   ...
}
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I know with arrays you can do LAST to stop the iterations.

last (and next and redo) affect loops, not arrays. Specifically,

while (and until) loops:

while (...) {
   ...
   last;
   ...
}
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C-style for loops, foreach loops and counting loops:

for (...) {
   ...
   last;
   ...
}
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Bare loops:

{
   ...
   last;
   ...
}
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See also perlsub