One way to get your desired output:

use warnings;
use strict;

my $line1;
my $line2;
while (<DATA>) {
    if (/^>/) {
        if ($line1) {
            print "$line1\n$line2\n";
            $line1 = '';
            $line2 = '';
        }
        print;
    }
    elsif (!/\./) {
        chomp;
        $line1 .= $_;
    }
    else {
        chomp;
        $line2 .= $_;
    }
}
print "$line1\n$line2\n" if $line1;

__DATA__
> gi|11SB_CUCMA Train|1 21
MARSSLFTFLCLAVFINGCLSQIEQQSPWEFQGS
EVWQQHRYQSPRACRLENLRAQDPVRLLLPGFSNAPKLIFV
AQGFGIRGIAIPGCAETYQT
SSSSSSSSSSSSSSSSSSSSS....................
...........................
.................
..........
> gi|1A43_HUMAN Train|1 24
MAVMAPRTLVLLLSGALALTQTWAGSHSMRYFYTSVSRPG
RGEPRFIAVGYVDDTQFVRFDSDAASQRMEPRAPWIEQEGPEYWSQTDRANLGTLRGYYNQSEDGSHTIQ
+R
MYGCDVGPDGRFLRGYQQDAYDGKDYIALNEDLRSWTAADMAAQITQRKWETAHEAE
SSSSSSSSSSSSSSSSSSSSSSSS..............................................
+...........................................
.............
..........................................
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