Your code has variables that can be (accidently!) used outside of the necessary scope. This is dangerous.

Yes, but it's good to realize that "use strict" doesn't magically fix that. Accidently using variables outside of the necessary scope happens when variables are defined using a scope that's too broad. It's pretty hard to accidently use variables from a different package - you would have to explicitely point to that package. Problems are cases like:

$var = "whatever";
sub foo {
    ...
    while ($var = <HANDLE>) {
        ...
    }
    ...
}
sub bar {
    ...
    foo();
    ... code assuming $var is unmodified ...
}
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Slapping a use strict; on top of the code, and putting a my in front of the first occurance of $var doesn't solve this problem - although it makes use strict very happy.

What does solve the problem is declaring a lexical variable my $var, or a local value local $var inside foo(), neither of which actually requires use strict.

use strict is a tool. It ain't no magical bullet.