Thats a very common problem, you are trying to add a record only when the successor is to be parsed, but the last record has no successor (sic ;)!

Most people try to solve by repeating code to add the last record after the loop.

But it's much cleaner this way (avoiding a posteriori state logic)

use strict;
use warnings;
use Data::Dump;

my $header;
my %sequence;

while ( my $line = <DATA> ){
    chomp $line; 
    if ( $line =~ /^>(.*)/ ) {
        $header = $1;
    } else {
        $sequence{$header} .= $line;
    }
}

dd \%sequence;

    
    
__DATA__
>sequence_5849
CCTGCGGAAGATCGGCACTAGAATAGCCAGAACCGTTTCTCTGAGGCTTCCGGCCTTCCC
TCCCACTAATAATTCTGAGG
>sequence_5959
CCATCGGTAGCGCATCCTTAGTCCAATTAAGTCCCTATCCAGGCGCTCCGCCGAAGGTCT
ATATCCATTTGTCAGCAGACACGC
>sequence_0808
CCACCCTCGTGGTATGGCTAGGCATTCAGGAACCGGAGAACGCTTCAGACCAGCCCGGAC
TGGGAACCTGCGGGCAGTAGGTGGAAT
[download]

output

{
  sequence_0808 => "CCACCCTCGTGGTATGGCTAGGCATTCAGGAACCGGAGAACGCTTCAGAC
+CAGCCCGGACTGGGAACCTGCGGGCAGTAGGTGGAAT",
  sequence_5849 => "CCTGCGGAAGATCGGCACTAGAATAGCCAGAACCGTTTCTCTGAGGCTTC
+CGGCCTTCCCTCCCACTAATAATTCTGAGG",
  sequence_5959 => "CCATCGGTAGCGCATCCTTAGTCCAATTAAGTCCCTATCCAGGCGCTCCG
+CCGAAGGTCTATATCCATTTGTCAGCAGACACGC",
}
[download]

update

a *general pattern* to solve such problems while staying DRY is to use references

    if ( $line =~ /^(HEAD_PATTERN)/ ) {
        $data    = \ $deeply{nested}{structure}{$1}; # reference data 
+       
    } else {
        $$data .= $line;                            # derefrence data
    }
[download]

like this you don't need to repeat the path of a deeply nested data structure, which might vary in multiple dimensions

update

added some explanation

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