No Cd, want to install to Portable HDD

So basically i have this old computer, with no cd drive or hdd. i want to install any flavor of linux on it. I have a portable hdd which i will put a 5gb partition on it for linux, so when im not using it, i can use it on my other comp for files.

the problem: BIOS doesnt support usb booting.

so this is what i need:
a way to install linux on the portable hdd. i dont care if it is a floppy network install, or if i can install it from my other computer.

and a floppy to boot onto the portable hdd

You could try Debian via floppy/network install:

Installing Debian GNU/Linux via the Internet