$. can only be used while still reading the file.

open my $LOG, '<', $logfile or die $!;
my %occurrences;
while (my $line = <$LOG>) {
    if ($line =~ /^$FAILURE_SEARCH /) {
        chomp $line;
        $occurrences{$.} = $line;
    }
}
return \%occurrences
[download]

When iterating over an array, you can use the index:

my %occurrences;
for my $index (0 .. $#log) {
    if ($log[$index] =~ /^$FAILURE_SEARCH /) {
        $occurrences{ 1 + $index } = $log[$index];
    }
}
return \%occurrences
[download]

Warning: untested code!

Update: Tested, fixed errors (missing argument to chomp, missing + 1 for the array index.