use warnings;
use strict;
use Data::Dumper;

my @lines = <DATA>;

my $n;
my %all_hash;
for (@lines) {
        chomp;
        if (/^>/) {
                $n = $_;
        } else {
                push @{ $all_hash{ $n } }, $_;
        }
}       
print Dumper(\%all_hash);

__DATA__
>seq1
ASDFGHASDFGHJ
ERTYUIOOIUYLK
NBGFEWERTY
>seq2
BGTNHYMJUKOPK
MNBFSDFGHJ
[download]

$VAR1 = {
          '>seq1' => [
                       'ASDFGHASDFGHJ',
                       'ERTYUIOOIUYLK',
                       'NBGFEWERTY'
                     ],
          '>seq2' => [
                       'BGTNHYMJUKOPK',
                       'MNBFSDFGHJ'
                     ]
        };
[download]

• Because your $all_hash{ $n } = $m; is inside the for loop, you clobber the prior instance each time you reach a new one.
• Your variable names make your life more difficult
• As written, the value of your $n includes the &lt; (<) sign. Do you really want that?

The following may be what you're seeking:

#!/usr/bin/perl
use warnings;
use strict;
use 5.012;

# 902200

my @lines = qw(
    >seq1
    ASDFGHASDFGHJ
    ERTYUIOOIUYLK
    NBGFEWERTY
    >seq2
    BGTNHYMJUKOPK
    MNBFSDFGHJ
    >seq3
    USE_STRICT&USE_WARNINGS
    lastline
);


my $line;
my ($DNA, $seq);            # descriptive var names
my %all_hash = ();
for my $line (@lines){      # insert after this line, see update 2
        chomp $line;
        if ($line =~ /^>(seq\d)/ ) {  # captures seq# withOUT the '>'
                $seq = "$1\n";
             say "\n\$seq: $seq";  # useful for debug; otherwise, not
        } else {
            
                $DNA = "$line\n";
            say "\$DNA: $DNA";
        }
        no warnings;    #otherwise, warns 'unitialized' for the first 
+$DNA
        $all_hash{ $seq } .= $DNA;    # concat $DNAs, which now 
                                      # have '\n's restored for readab
+ility
        use warnings;
}

print "\n =============== \n";

print %all_hash;
[download]

Output (when fixed as per update 2):

$seq: seq1

$DNA: ASDFGHASDFGHJ

$DNA: ERTYUIOOIUYLK

$DNA: NBGFEWERTY


$seq: seq2

$DNA: BGTNHYMJUKOPK

$DNA: MNBFSDFGHJ


$seq: seq3

$DNA: USE_STRICT&USE_WARNINGS

$DNA: lastline


 ===============
seq1
ASDFGHASDFGHJ
ERTYUIOOIUYLK
NBGFEWERTY
seq3
USE_STRICT&USE_WARNINGS
lastline
seq2
BGTNHYMJUKOPK
MNBFSDFGHJ
[download]

Update

Update2 : Missed a serious mistake here: the concat at line 36 actually inserts the last $DNA from the previous $seq into each new $seq. Bad on me! A fix is to reset $DNA to an empty string by inserting $DNA = ''; as a new line 26.