Simple. Just open a file with the filehandle of STDERR:

#!/usr/bin/perl

use strict;

open(STDERR,">>error.log") || die "Couldn't redirect STDERR: $!\n";
print STDERR "This goes to the file.\n";
warn "As does this.";
die "And this does, too!\n";
close STDERR;
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Update: If you want to restore the original filehandle, then you'll have to do something like this:

open(OLDERR,">&STDERR") || die "Couldn't dup STDERR: $!\n";
open(STDERR,">>error.log") || die "Couldn't redirect STDERR: $!\n";
print STDERR "This goes to the file.\n";
warn "As does this.";
open(STDERR,">&OLDERR") || die "Couldn't restore STDERR: $!\n";
warn "This goes to the screen.";
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Btw, this is almost entirely taken from perldoc -f open.

mr.nick ...