Again, please use <code> tags and read How do I post a question effectively?

I have found strict to be very helpful to identifying and avoiding bugs in my code -- see Use strict warnings and diagnostics or die. If I were going to write your posted code, it might look more like:

use strict;
use warnings;

open (my $out, ">", "Results.txt") or die ("Could not open file Result
+s.txt; $!");
open (my $in, "<", "Textfile.txt") or die ("Can not open input file: $
+!");
local $/;
while (<$in>) {
    if (/(web site.{250})/i) {
        print $out $1;
    }
}
[download]

Changes that I made include:

1. I swapped to lexical file handles and 3 argument open, which are considered better practice for a number of reasons. See perlopentut. In particular, this gives strict more power to help and removes the need for explicit close.
2. I corrected inconsistency between your file name and error message; file names are generally case sensitive.
3. I swapped to slurp mode using $/. Given the large number of characters you are interested in, it is unlikely they will all fall on the same line.
4. Your while(<>) loop read data into $_ not $ARG, so I corrected that.
5. I swapped your regular expression to the regular expression I posted above, with the addition of the s modifier. This makes it so . also matches new lines, and is essential when working in slurp mode.

You may consider going to http://learn.perl.org to gather some learning resources before trying to run too far.