otherwise its value doesn't mean anything at all.

Isn't "at all" too strong here? $! (i.e. errno is initially zero (as far I remember, this is guaranteed by the C standard), and gets a non-zero value if someone sets it - for instance, in a failed system call or if in Perl it is explicitly set by assignment. Seeing it only this way, the value of $! picked up somewhere in the middle of a program execution indeed doesn't mean much.

But here we have a different situation: We see that in the program

perl -lwe "print $!"
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$! is zero, which means that when Perl starts executing a (correct) program, $! happens to be zero. We also see that for instance

perl -lwe "use warnings; print $!"
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print zero, so just the fact that Perl has to compile a pragma, doesn't change the value of $!. But we see that

perl -lwe "use strict; print $!"
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prints a non-zero value, and from this I conclude that $! must be set somewhere somehow within strict.pm. Being a curious person, I had a look at the source of strict.pm. I didn't find a direct assignment to $! (which is no surprise - we wouldn't expect this anyway), and the system calls I found (and which could possibly set $! to a 'bad file descriptor' error number), are in blocks belonging to error handling (i.e. I would have seen an error message, if such a branch would have taken). That's why I was puzzled and became curious. I admit that it is only for academic interest and has not much practical value aside of satisfying my curiosity ;-)