Here is your code modified in a few ways:

• There's no need for a main() sub here so I've removed that. The script will run fine without it.
• The @js array is no longer needed so has been removed.
• open now uses the 3-argument form and a lexical filehandle. If it fails the failure message now says why (it could easily be some reason other than the file not being found).
• The results are now stored in a hash since you mentioned you were after a unique list.
• I've had to assume the format of your input data because you never showed that.
• Lots of the places you've used brackets where Perl doesn't require them so those have been removed for clarity.
• I've used postfix versions of unless and for to reduce the number of indented blocks.

The rest of your variable names have been left the same. If I were writing this myself I would lowercase them all but that's purely a stylistic thing.

#!/usr/bin/env perl 
use strict;
use warnings;

my %ujs;
my $file = 'access_log.txt';
open my $FH, '<', $file or die "File $file not opened: $!";

while (my $String = <$FH>)
{
    next unless $String =~ /([^\/]*\.js)/;
    $ujs{$1} = 1;
}

close $FH;
print "$_\n" for sort keys %ujs;
[download]

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