You need to: 1)remember the $fname across loops, so define it outside; 2)also you need to my %hoa; 3) there is no need to define that $content.

use Data::Dumper;

use strict;
use warnings;

my %hoa;
my $fname;
while (my $line = <DATA>) {
    chomp $line;
    if ($line =~ /^[a-z]/ )
    {
        $fname = $line;
    }
    elsif ($line =~ /^[0-9]+,/)
    {
        push @{$hoa{$fname}}, $line;
    }
}               

print Dumper(\%hoa);

__DATA__
>data set
dm01
>instances
0,-869,foofoofoofoofoofoofoofoo,28
0,-853,barbarbarbarbar,14
>data set
yst02
>instances
0,-1566,quxquxquxqux,9
0,-1545,bembembem,9
0,-1394,birbirbirbir,9
>data set
mus03
>instances
0,-1274,tingtingtingting,11
0,-1220,tongtongtong,11
0,-475,bubububu,11
0,-459,catcatcatcatcat,11
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