I would eliminate the use of chdir. Yes, the program as given works, but only because both of the directories are specified as absolute. I think you'd gain flexibility by allowing relative paths (for example, "." as the srcdir).

Here's how I might rewrite it:

#!/usr/bin/perl

use Getopt::Long;
use strict;
use warnings;
$|=1;

my $srcdir;
my $destdir;
my $mtime = 1;

GetOptions(
        'srcdir=s' => \$srcdir,
        'destdir=s' => \$destdir,
        'mtime|days=i' => \$mtime,
);

defined $srcdir && defined $destdir or die <<EOF;
Usage: $0 --srcdir SRCDIR --destdir DESTDIR [-days N]
EOF

opendir SRCDIR, $srcdir or die "opendir $srcdir - $!";

my @too_old;
my @files;

# only select non-hidden regular files:
for ( grep { -f "$srcdir/$_" } readdir SRCDIR )
{
        if ( -M "$srcdir/$_" > $mtime )
        {
                push @too_old, $_;
        }
        else
        {
                push @files, $_;
        }
}

closedir SRCDIR;

@too_old and
print "(Ignoring ".scalar(@too_old)." too-old files.)\n";

@files or die "No backup candidates in $srcdir.";

unless ( -d $destdir )
{
        warn "Destination directory $destdir does not exist; attemptin
+g to create.\n";
        mkdir $destdir or die "mkdir $destdir - $!";
}

print "Backing up ".scalar(@files)." files from $srcdir.\n";

for my $f ( @files )
{
        print "Archiving $f \n";
        system qq( cd "$srcdir" ; tar -cz "$f" -f "$f.tar.gz" );
        system qq( mv "$srcdir/$f.tar.gz" "$destdir" );
}
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This doesn't entirely avoid chdir, but it puts it at the lowest, most constrained context possible. In fact, if you have gnu tar (which it appears you do), you can exploit the -C option to make tar itself do the directory change within itself.