The first problem is that your $word_1 has a newline at the end. Since this is almost surely not what you want to do, you should chomp both words after you read them in.

As BazB has pointed out above, you're opening and closing the file several times unnecessarily. Also, if your open fails, there's not much clue about what's happening. It's much clearer to check for errors when opening, then close the handle explicitly, then open it for writing (since you'll be re-writing its contents).

Finally, if you're going to read in the whole file and do match-and-substitute, it's a lot easier not to use an array. Just read the whole thing into a scalar, and do a global replace. Put it all together, and it looks like this:

open HANDLE, $file
    or die "Couldn't open $file: $!\n";
print "Please Enter a word from $file: ";
chomp($word_1 = <STDIN>);
print "Please Enter a Word to Replace, $word_1, with: ";
chomp($word_2 = <STDIN>);

# Read the whole thing in.
undef $/;
$x = <HANDLE>;
close HANDLE;

# Re-open for writing, clobbering the file.
open HANDLE, ">$file";
# Change every instance
$x =~ s/$word_1/$word_2/g;
# To prevent changing substrings, it would almost
# certainly be better to use:
#    $x =~ s/\b$word_1\b/$word_2/g;
# But it's your call....
print HANDLE $x;
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At this point, I should probably also chide you for not using strict and warnings, but I'll let you off easy this time. ;-)

HTH