next unless {-d "$ldir/$name"};
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   next unless -d "$ldir/$name";
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Also, I think you should change:

 my @races = undef;
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to:

my @races;
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Tip #4 from the Basic debugging checklist: After your 1st while loop, add:

use Data::Dumper;
print Dumper(\@races);
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You've a typo in the line:

Since you use single quotes $race and $i are not substituted by their value. Just try:

My anonymous brother is quite right. You can avoid this sort of problem (where by you test one thing but actually report another) by using a variable name for both. eg.

      my $file = "/strek/$race/logs/rlog$i.sr.html";
      if ( -s $file ) {
         next;
      } else {
         print "$file either does not exist or is 0 bytes!\n";
      }
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Other things you could do include making more use of the $ldir variable and rolling the second loop into the first.

#!/usr/bin/perl

use strict;
use warnings;

my $ldir = '/strek';
my @races = undef;

opendir my $dh, $ldir
   or die "$0: opendir: $!";

while (defined(my $name = readdir $dh)) {
   next unless -d "$ldir/$name";
   push (@races, $name);
   for my $i (0..4) {
      my $file = "$ldir/$name/logs/rlog$i.sr.html";
      if ( -s $file ) {
         next;
      } else {
         print "$file either does not exist or is 0 bytes!\n";
      }
   }
}

# Now do something useful with @races here
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while (my $race = <@races>) {
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foreach my $race (@races) {
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#!/usr/bin/perl 
use strict;
use warnings;

my $ldir = '/strek';
my @races = undef;

opendir my $dh, $ldir
   or die "$0: opendir: $!";

while (defined(my $name = readdir $dh)) {
   next unless {-d "$ldir/$name"};
   push (@races, $name);
}

for (@races) {
   for my $i (0..4) {
      if ( -s "/strek/$_/logs/rlog$i.sr.html" ) {
         
      } else {
         print "/strek/$_/logs/rlog$i.sr.html either does not exist or
+ is 0 bytes!\n";
      }
   }
}
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You don't want to initialize your array like so: my @races = undef; because like so, you already have an element in the array. Which is not what you want to do.
You can simply do my @races = (); OR more better my @races; just like toolic posted above.

If you tell me, I'll forget.
If you show me, I'll remember.
if you involve me, I'll understand.
--- Author unknown to me

Well, does this work?

>perl -wMstrict -le
"opendir my $dh, '.' or die qq{opening .: $!};
 while (defined(my $dname = readdir $dh)) {
   next unless { -d $dname };
   print qq{dir '$dname'};
   }
"
Odd number of elements in anonymous hash at -e line 1.
dir '.'
Odd number of elements in anonymous hash at -e line 1.
dir '..'
Odd number of elements in anonymous hash at -e line 1.
dir '10001fr.equ'
Odd number of elements in anonymous hash at -e line 1.
dir '10002fr.equ'
...
Odd number of elements in anonymous hash at -e line 1.
dir 'Wrap_Simple.pm'
Odd number of elements in anonymous hash at -e line 1.
dir 'Wrap_Simple.pm.bak'
Odd number of elements in anonymous hash at -e line 1.
dir 'xxxx'
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for my $dir (glob "/strek/*/logs") {
    for my $i (0 .. 4) {
        print "$dir/rlog$i.sr.html either does not exist or is 0 bytes
+!\n"
            unless -s "$dir/rlog$i.sr.html";
    }    
}
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while (my $race = <@races>)

for my $race (@races)

You have been given a lot of useful comments on various faulty details in your code, but I was going to suggest the same thing as Cristoforo: I think that you should take a look at the glob function. It is often much more practical in a number of respects than opendir/readdir.