It really sound as if the hard drive has been damaged. In an attempt to save data, it might be possible to open the drive, as Jim has indicated, but buying a new unit is probably going to be unavoidable.
Separate image files are normally used to create individual floppy disks, via a suitable disk image program (such as Disk Copy 4.2 or 6.3.3 on older Macs). The downloadable System 7.5.3 on the other hand has 19 parts, forming one large disk image (Disk Copy is not required here). Just copying image files to a ZIP disk is not going to help. You will have to either (via an installer) install a minimal or full system onto the ZIP disk, or drag-copy an existing (and appropriate) system folder to the ZIP disk. The problem is that the ZIP disk, with the thus installed or transferred system, might not become bootable unless another older Mac is used.
The best solution at this point is to ask someone with such an older Mac for assistance. The first step could then be to create a bootable floppy (e.g. the Network Access disk mentioned earlier, or a full set of system floppies from disk image files). Please note the Color Classic operating system requirements (see the technical specifications
here). Check whether you can start the computer from a floppy. If necessary, replace the internal hard drive and install a fresh operating system.
You can connect the Color Classic to another old Mac via a plain Macintosh serial printer cable (MiniDIN-8M to MiniDIN 8-M). If the Color Classic has been started from a Network Access floppy, files on the other machine can be shared via AppleTalk.
Jan