# Taken from The Camel, 3rd ed.
while (<>) {
  chomp;
  if (s/\\//) {
    $_ .= <>;
    redo unless eof;    # don't read past each file's eof
  }
  # now proces $_      
}

##</code><code>##

# file_1 (this line doesn't belong to the file's content)
Line one\
Line two

##</code><code>##

# file_2 (this line doesn't belong to the file's content)
Line three
Line four\
# file_3 (this line doesn't belong to the file's content)
Line five
Line six

##</code><code>##

# My code
while (<>) {
  chomp;
  if (s/\\//) {
    unless (eof) {    # don't read past each file's eof
      $_ .= <>;
      redo;
    }
  }
  # now process $_
}