Copying as a matter of using the software or making archives of the software is legal.[1] There is also no practical way to police what happens and never leaves the inside of your home.
Distribution of copied software on the other hand is very illegal, and is what most so-called "archivists" (including the Internet "Archive") run afoul.
Internet Archive would be having no legal issues if they simply copied software they have a right to possess and then stored the copies in their own servers inaccessible to anyone but themselves, also known as archiving.
And no, archives do not need to be publically accessible.