The main problem with Windows Explorer is that it doesn't support resuming copying after a read error. Most superficially scratched CDs, for example, will fail on different areas of the disk every time you eject and reinsert them.
Therefore, with a utility that supports resuming copy operations, it is possible to read the entire contents of a damaged CD with by doing "eject/reload/resume" a few times.
In fact, this is what a utility I wrote does, and I've never needed anything fancier to read scratched disks. (It simply uses ReadFile and WriteFile.)
One step lower would be opening the raw partition (i.e. disk image) by passing a string such as "\.\F:" (note: slashes are literal here) to CreateFile. It would allow you to read raw sectors from a drive, but reconstructing files from that data would be hard.
In fact, the "\.\" syntax allows you to open devices in the "\GLOBAL??" branch of the Windows Object Manager namespace as if they were files. It's not unlike calling dd with /dev/x as a parameter. There is also a "\Device" branch, but that's only accessible via DeviceIoControl() (i.e. ioctl()), meaning there's no simple ReadFile()/WriteFile() interface.
Anything lower level than that would be device-specific, I guess; like reading raw CD-ROM data (including ECC bits) the way some CD-burning programs do. You'd have to do some research on the specific media (CD, flash, DVD) and what your hardware allows you to do on them.
Note: The backslashes seem to get lost on the way to the web page; you need to pass "backslash backslash dot backslash DeviceName" to CreateFile. You need to escape them, too, of course.