update: changed chop to chomp on Aighearach's suggestion, which is of course very sensible, as one doesn't know for sure that the record separator will only be one character.

AFAIK you have to read through the file. You can only pull out a pre-determined fragment of the file if you know its location in bytes. Then you would use seek and read. But unless it's a huge file, there's no harm in doing something like

my $start = 'banana';
my $end = 'grape';
my $get = 0;
my @results;

open FILE, "file.txt";
while (<FILE>) {
    chomp;
    $get = 0 if $_ eq $end;
    push @results, $_ if $get;
    $get = 1 if $_ eq $start;
}

print $_,"\n" for @results;
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You'd need something a little more subtle if you aren't sure that 'banana' comes before 'grape' or you think one or other might be absent altogether.

§ George Sherston