I don't know of an existing module but you could do it something like this:

sub get_nice_chunk {
  my($fh, $size) = @_;

  local $/ = '';
  local $_;
  my $chunk;

  $chunk .= $_
    while $_ = <$fh> and length($_) + length($chunk) < $size;

  seek $fh, tell($fh) - length($_), 0
    if length($_) + length($chunk) < $size;

  return $chunk;
}

print "got chunk: $_"
  while $_ = get_nice_chunk(\*DATA, 19);

__DATA__
foo
bar
baz
quux

ichi
ni
san
shi

the
last
bit
[download]

So that just sets the INPUT_RECORD_SEPARATOR to paragraph mode, reads paragraphs until we've hit the $size limit, rewinds the file pointer and returns the chunk. However, this does suffer from the bug that it won't read pargraphs greater than $size but I'll leave that as an exercise to the OP of what to do in that case.