The general rule when it comes to parsing character separated data is: Don't. Use a module instead. In this case Text::xSV is most likely appropriate. Consider:

use strict;
use warnings;
use Text::xSV;

my $xsv = Text::xSV->new (sep => "\t", fh => *DATA);

while (my @row = $xsv->get_row ()) {
    map {s/\s+$//; s/^\s+//} @row;
    print ">", join ("< >", @row), "<\n";
}

__DATA__
   0.000      12    0.232    13    11      text that can have space
   1.000      13    0.534    14    12      More text that would be ok
   2.000      14    0.876    15    13      yet more text
[download]

Prints:

>0.000< >12< >0.232< >13< >11< >text that can have space<
>1.000< >13< >0.534< >14< >12< >More text that would be ok<
>2.000< >14< >0.876< >15< >13< >yet more text<
[download]

Update: altered code to use __DATA__ rather than heredoc.