Whenever you find yourself copying and pasting code, or generating a bunch of lines that look similar either use a sub or consider using an appropriate data structure. The following rework of your code checks opens, uses the three parameter version of open, uses lexical file handles, "fixes" the grep of a single element issue, cleans up the binning code (do you really want unequal bin sizes?) and adds some test data along with a file reading variant that uses it.

use strict;
use warnings;

my ($fastaFile, $rsite1, $rsite2, $rsite3, $organism) = @ARGV;
my %genome = fastaReadData();
my $radtagfile = join '_', $organism, "Radseq_fragments_doubleDigest",
    $rsite1,
    $rsite2, "${rsite3}fasta";
my $sizesfile = join '_', $organism, "Radseq_fragments_doubleDigest",
    $rsite1,
    $rsite2, $rsite3, "sizes.txt";

open my $fOut,  '>', $radtagfile or die "Can't create $radtagfile: $!\
+n";
open my $fSize, '>', $sizesfile  or die "Can't create $sizesfile: $!\n
+";
print $fSize "length\torganism\n";

# prepare a summary file
my ($Second, $Minute, $Hour, $Day, $Month, $Year) = localtime(time);

$Month++;
$Year += 1900;

my $date = join '_', $Month, $Day, $Year;
my $summaryfile = join '_', $organism, "summary_doubleDigest", $rsite1
+,
    $rsite2,
    $rsite3, "${date}txt";

open my $fRes, '>', $summaryfile or die "Can't create $summaryfile: $!
+\n";

my %final_fragments;
my @all_size_fragments;

# start creating the fragments
foreach my $seqname (keys %genome) {
    print "Working on sequence $seqname.\n";

    # split the fragments up based on the enzyme motif
    my @first_fragments = split $rsite1, $genome{$seqname};

    # add the restriction site motif back onto the fragments
    $first_fragments[0] .= $rsite1;
    $first_fragments[-1] = "$rsite1$first_fragments[-1]";

    for my $i (0 .. $#first_fragments) {
        if ($i or $i != $#first_fragments) {
            $first_fragments[$i] = "$rsite1$first_fragments[$i]$rsite1
+";
        }

        # now split the fragment using $rsite2
        # repair the first and last fragments to include $rsite2
        # these are the only fragments to contain both restriction sit
+es, so
        # keep them in @final_fragments
        my @second_fragments = split $rsite2, $first_fragments[$i];

        $second_fragments[0] .= $rsite2;
        $second_fragments[-1] = "$rsite2$second_fragments[-1]";
        push @all_size_fragments, length $_ for @second_fragments;

        my @third_fragments = grep !/$rsite3/, @second_fragments;

        $third_fragments[0] .= $rsite3;
        $third_fragments[-1] = "$rsite3$third_fragments[-1]";
        push @all_size_fragments, length $_ for @third_fragments;

        $final_fragments{join '_', $seqname, $i, 1} = $third_fragments
+[0];
        $final_fragments{join '_', $seqname, $i, 2} = $third_fragments
+[-1];
    }

}

# keep a score of how many fragments fall within a particular size ran
+ge

my %counts;
my @bins = (('smaller than 100') x 2, 'between 100 and 150');

push @bins,
    map {'between ' . ($_ * 50 + 1) . ' and ' . ($_ + 1) * 50} 3 .. 12
+;
push @bins, 'larger than 600';
$counts{$_} = 0 for @bins;

my $totalBP = 0;  # the total number of base pairs in the selected fra
+gments

foreach my $fragment (keys %final_fragments) {
    # add on $rsite1 to both sides of the fragment
    my $fragmentLength = length($final_fragments{$fragment});

    print $fOut ">${fragment}_1\n";
    print $fOut substr($final_fragments{$fragment}, 0, 96), "\n";
    print $fOut ">${fragment}_2rc\n";
    print $fOut revcom(
        substr($final_fragments{$fragment}, $fragmentLength - 96, 96)
        ),
        "\n";
    $totalBP += $fragmentLength;

    my $binnedSize = $fragmentLength / 50;

    $binnedSize = 2 if $binnedSize < 2 && $fragmentLength >= 100;

    if ($binnedSize >= $#bins) {
        ++$counts{$bins[-1]};
    } else {
        ++$counts{$bins[$binnedSize]};
    }
}

my $totalCount = keys %final_fragments;
print $fRes "The restriction sites used were:\n";
print $fRes "$rsite1\n";
print $fRes "$rsite2\n\n";
print $fRes "There were ", $totalCount,
    " fragments from the whole genome.\n";
print $fRes "There were $counts{$_} fragments $_ bp.\n"
    for @bins[1 .. $#bins];
print $fRes "There are ", $totalBP, " base pairs in the fragments.\n";
print $fRes "\nJust some notes of reference:\n";
print $fRes "HiSeq 2000 gives 375,000,000 reads.\n";
print $fRes "HiSeq 2500 gives 742,000,000 reads.\n";

close $fOut;
close $fRes;

print $fSize "$_\t$organism\n" for @all_size_fragments;


sub fastaReadData {
    return parseFile(*DATA);
}


sub fasta_read_gzip_alt {
    my ($filename) = @_;    # be sure to include the path
    my %fasta;
    open my $fIn, "gunzip -c $filename |"
        or die "can't open pipe to $filename: $!";
    return parseFile($fIn);
}


sub fasta_read_alt {
    my ($filename) = @_;    # be sure to include the path

    open my $fIn, '<', $filename or die "can't open $filename: $!\n";
    return parseFile($fIn);
}


sub parseFile {
    my ($fIn) = @_;
    my $fastaData;
    my $sequence = '';
    my $name     = '';
    my %fasta;

    while (<$fIn>) {
        $fastaData = $_;
        $fastaData =~ s/\n//gms;
        if ($fastaData =~ />/) {
            if ($sequence) {
                # if there is a sequence, then the sequence belongs to
+ the last
                # name
                $fasta{$name} = $sequence;
            }

            # reinitialize everything
            $sequence = '';           # start over!
            $name     = $fastaData;
            $name =~ s/>//gms;
        } elsif (eof $fIn) {
            $fasta{$name} = $sequence;
        } else {
            $sequence .= $fastaData;
        }
    }

    return %fasta;
}


sub revcom {
    (my $sequence) = @_;
    $sequence = reverse($sequence);
    $sequence =~
        tr/AGCTRYMKSWHBVDNagctrymkswhbvdn/TCGAYRKMSWDVBHNtcgayrkmswdvb
+hn/;
    return $sequence;
}


__DATA__
>24.6jsd1.Tut
TTGGAGAGTTTGATCCTGGCTCAGGATGAACGCTGGCGGCGTGCCTAATA
CATGCAAGTCGAGCGAATGGATTAAGAGCTTGCTCTTATGAAGTTAGCGG
CGGACGGGTGAGTAACACGTGGGTAACCTGCCCATAAGACTGGGATAACT
CCGGGAAACCGGGGCTAATACCGGATAACATTTTGAACCGCATGGTTCGA
AATTGAAAGGCGGCTTCGGTCGTCACTTATGGATGGACCCGCGTCGCATT
AGCTAGTTGGTGAGGTAACGGCTCACCAAGGCAACGATGCGTAGCCGACC
TGAGAGGGTGATCGGCCACACTGGGACTGAGACACGGCCCAGACTCCTAC
GGGAGGCAGCAGTAGGGAATCTTCCGCAATGGACGAAAGTCTGACGGAGC
AACGCCGCGTGAGTGATGAAGGCTTTCGGGTCGTAAAACTCTGTTGTTAG
GGAAGAACAAGTGCTAGTTGAATAAGCTGGCACCTTGACGGTACCTAACC
AGAAAGCCACGGCTAACTACGTGCCAGCAGCCGCGGTAATACGTAGGTGG
CAAGCGTTATCCGGAATTATTGGGCGTAAAGAACGCGCAGGTGGTTTCTT
AAGTCTGATGTGAAAGCCCACGGCTCAACCGTGGAGGGTCATTGGAAACT
GGGAGACTTGAGTGCAGAAGAGGAAAGTGGAATTCCATGTGTAGCGGTGA
AATGCGTAGAGATATGGAGGAACACCAGTGGCCCAGGCGACTTTCTGGTC
TGTAACTGACACTGAGGCGCGAAAGCGTGGGGAGCAAACAGGATTAGATA
CCCTGGTAGTCCACGCCGTAAACGATGAGTGCTAAGTGTTAGAGGGTTTC
CGCCCTTTAGTGCTGAAGTTAAAGCATTAAGCACTCCGCGTGTGGAGTAC
GGCCGCAAGGCTGAAACTCAAAGGAATTGACGGGGGCCCGCACAAGCGGT
GGAGCATGTGGTTTAATTCGAAGCAACGCGAAGAACCTTACCAGGTCTTG
ACATCCTCTGACAACCCTAGAGATAGGGCTTCTCCTTCGGGAGCAGAGTG
ACAGGTGGTGCATGGTTGTCGTCAGCTCGTGTCGTGAGATGTTGGGTTAA
GTCCCGCAACGAGCGCAACCCTTGATCTTAGTTGCCATCATTWAGTTGGG
CACTCTAAGGTGACTGCCGGTGACAAACCGGAGGAAGGTGGGGATGACGT
CAAATCATCATGCCCCTTATGACCTGGGCTACACACGTGCTACAATGGAC
GGTACAAAGAGCTGCAAGACCGCGAGGTGGAGCTAATCTCATAAAACCGT
TCTCAGTTCGGATTGTAGGCTGCAACTCGCCTACATGAAGCTGGAATCGC
TAGTAATCGCGGATCAGCATGCCGCGGTGAATACGTTCCCGGGCCTTGTA
CACACCGCCCGTCACACCACGAGAGTTTGTAACACCCGAAGTCGGTGGGG
TAACCTTTTTGGAGCCAGCCGCCTAAGGTGGGACAGATGATTGGGGTGAA
GTCGTAACAAGGTAGCCGTATCGGAAGGTGCGGCTGGATCACCTCCTTTC
T
[download]

The summary file contains:

The restriction sites used were:
TTGG
CTAG

There were 18 fragments from the whole genome.
There were 14 fragments smaller than 100 bp.
There were 3 fragments between 100 and 150 bp.
There were 0 fragments between 151 and 200 bp.
There were 1 fragments between 201 and 250 bp.
There were 0 fragments between 251 and 300 bp.
There were 0 fragments between 301 and 350 bp.
There were 0 fragments between 351 and 400 bp.
There were 0 fragments between 401 and 450 bp.
There were 0 fragments between 451 and 500 bp.
There were 0 fragments between 501 and 550 bp.
There were 0 fragments between 551 and 600 bp.
There were 0 fragments between 601 and 650 bp.
There were 0 fragments larger than 600 bp.
There are 1121 base pairs in the fragments.

Just some notes of reference:
HiSeq 2000 gives 375,000,000 reads.
HiSeq 2500 gives 742,000,000 reads.
[download]

and no errors or warnings are raised.