Ok, I looked this up with Google:
Is your machine like this?
Pentium 75MHz Processor
340MB Hard Drive
Note this from the instructions:
minimum hardware you need in order to run ZipSlack.
* 386 processor
* 16 megabytes of RAM
* 100 megabytes of hard disk space on a FAT or FAT32 filesystem
NOTE: The additional RAM is needed to compensate for the generic kernel included in ZipSlack. There is an addon for ZipSlack which installs an 8 MB swap file in the \LINUX directory. You must have this addon if you want to boot ZipSlack on a system with less than 8 MB of RAM.
So no CDROM. Information like this is required to get a real answer.
ZipSlack is made to go on a Windows partition, so that you do not have to reformat your hard drive. I assume this box has Win 95 on it. The instructions are pretty clear. Copy the ZipSlack.zip file to your hard drive somewhere, then do this, from the instructions:
Use an unzip utility to unzip the file.
When the program asks you to select a destination to save the extracted files to, give it the drive letter of the disk you are installing to followed by a backslash. For example, if you are installing to C drive, tell the unzip program to extract to C:\. The unzip program should extract the files in the archive and save them to the disk. It will create a \LINUX subdirectory on the disk.
It is very important to extract ZipSlack to the top level directory of whatever drive you decide to install it to.
You will have to read the boot instructions to actuly run it.
FWIW, this is going to be a great learning experience.