chorobas solution and commentary is excellent; I don't disagree with any of it. I offer this as an alternate possible solution. (I was doing this before I saw that was posted.) Here I slurp the file in by paragraphs then pick out the lines I want to save.

#!usr/bin/perl
use strict;
use warnings;
use Tk;
use Cwd;
use Data::Dumper;
#
######################################################################
+#####################
# GUI Building 
######################################################################
+#####################
#
# Create Main Window
my $mw = MainWindow->new;

my $n = 0;
my %temp;

my $ms_button = $mw->Button(
    -text=>"MS",
    -command=> \&BJSFM_MS
)->pack();

MainLoop;

sub BJSFM_MS {
    my $filename = 'BJSFM_out.prn';
    #open my $infile, '<', $filename or die "Can't open $filename! $!"
+;
    my $infile = *DATA;
    local $/="FAILURE CRITERIA PER PLY\n";

    while (my $paragraph = <$infile>) {
                chomp $paragraph;
                $paragraph =~ s/^\s+\n|\s+$//g;
                my @lines = split "\n", $paragraph;
                next if @lines < 4;
                map {$temp{$n++} = $_ } @lines[3..$#lines];
        }
    close $infile;
    print Dumper \%temp;
}

__DATA__
                   FAILURE CRITERIA PER PLY

   DIST     ANGLE      PLY            FAILURE NUMBERS
                                   1         2       SHEAR
  0.000      0.00    -45.00     0.238     0.282    -1.459
  0.000      0.00      0.00     0.971     1.369     0.004
  0.000      5.00    -45.00     0.475     0.142    -1.585
  0.000      5.00      0.00     1.003     1.531    -0.274
  0.000     10.00    -45.00     0.721     0.037    -1.623

                   FAILURE CRITERIA PER PLY

   DIST     ANGLE      PLY            FAILURE NUMBERS
                                   1         2       SHEAR
  0.000      0.00    -45.00     0.247     0.293    -1.514
  0.000      0.00      0.00     1.008     1.422     0.004
  0.000      5.00    -45.00     0.493     0.147    -1.645
  0.000      5.00      0.00     1.042     1.589    -0.284
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