Your code has a number of problems. First of all, $str is never set back to nothing, so it eventually contains all data lines - not just the current section of data. Next, you seem to be assigning $str contents to $hash{$1} before actually putting anything into $str. Did you want to assign a reference instead? That would be $hash{$1} = \$str, and for that to work properly, $str would have to be a variable created fresh for each assignment to a new $hash{$1} - probably by doing an if statement with my $str inside it.

Basically, the code needs to be rewritten somewhat. Here's my version for what you're trying to do:

use strict;
use warnings;

my %hash;
while (<DATA>) {
    if (/>([\w]+) /) { }
    else { chomp; $hash{$1} .= $_; }
}
for (sort keys %hash) {
    print "$_ => $hash{$_}\n";
}

__DATA__
>EP11110 (-) 
TGCAATCACTAGCAAGCTCTC
GCTGCCGTCACTAGCCTGTGG
>EP40005 (+) 
GGGGCTAGGGTTAGTTCTGGA
NNNNNNNNNNNNNNNNNNNNN
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