I wonder if there is anyway I can make this system boot.
Easy. Turn on power. Done. The system should start, run its firmware (probably BIOS given the age of the system) and try to load the bootloader from whichever target is configured in the BIOS.
Now, without graphics this is where it get tricky. Most BIOS implementations on consumer motherboards need an attached monitor to help you configure them. So you probably want some graphics card just to set which device to boot as well to check what it will do on an 'error'. Some firmware consider missing keyboards or missing graphics out an error and have an option to either halt with an error message on the screen, or to continue regardless.
If this is the case with your setup thaen it gets interesting.
If not, simply put in a harddisk and a USB pendrive or a CDROM and try to install. If you have the another computer you could use that to look up which keys to press at which time. (VMs come in really handy here).
Alternatively, you can install an OS on a different computer and then just move the harddisk to the old PC. If your install BSD or Linux or similar, that should just work. If you want to install windows you might want to read up on the many posts here on SuperUser which mention sysprep.
Suppose that I have an Ethernet cable and an existing Windows laptop.
Will network boot work in this case?
It can. PXE booting is quite common and in theory nothing stop you from installing a PXE server on the laptop and trying to boot off that. In practice this might be a bit harder than just using a CD because you also need to configure a second system to serve the bootimage. And PXE booting might need to be enabled in the old computers BIOS. Technically though nothing is stopping you from doing that.
Still, the easiest way is to just install the OS on the SATA drive from a system with working graphics. E.g. connect it to your laptops eSATA port (assuming it has one) and do a windows install. Or a Linux install, or ...