I agree with Cere above - you are VERY likely NOT to have any problem.
The scary stories you have heard are rare, but a couple of ways MIGHT have made them happen. As Cere said, although the computer is not running, its PSU is still connected to wall power. In RARE circumstances a power line voltage surge could damage the PSU's input circuits, causing the computer not to work. Even then, there is NOT likely any damage beyond the PSU part. The other situation, also rare, is if the small BIOS configuration backup battery is worn out. This is a little silver disk about the size of a quarter coin sitting in a plasctic holder on the mobo. Its sole purpose is to keep alive the data stored in a special RAM chip associated with the mobo BIOS chip,, and to keep the clock running when the computer is NOT on. That battery normally lasts for years, but eventually it does wear out and needs replacement. When that happens, the computer WILL boot up, but then has lost its memory of which hard drive to boot from and gives a message hat you need to insert a bootable disk. For many users, the is a confusing message that causes panic, but there are simple ways to fix that, too.
Bottom line, neither of these is likely to happen to you, so do not worry.