You want to use time, not gmtime, and use strftime( localtime( $start )) to turn the seconds back into strings.

That gets us there:

$ ./2.strftime.pl 
It took me 10 seconds
I was started on 2019-12-03 11:41:38 and ended on 2019-12-03 11:41:48
$ cat 2.strftime.pl
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#!/usr/bin/perl
use strict;
use warnings;
use POSIX qw(strftime);
my $start = time();
sleep 10;
my $end      = time();
my $duration = $end - $start;
print "It took me $duration seconds\n";
my $start_date = strftime "%Y-%m-%d %H:%M:%S",(localtime($start));
my $end_date   = strftime "%Y-%m-%d %H:%M:%S",(localtime($end));
print "I was started on $start_date and ended on $end_date\n";
[download]

I'll fix the above post.

Fixing a post from 2010 might have a butterfly effect. Getting better behavior from output is what happens with code review and replication in "the monastery."