Purchase Server Components

Below is a list of things you'll need if you are going to build your server piece by piece:

CPU
Motherboard Compatible with CPU
DRAM Compatible with Motherboard
Case in same form factor as Motherboard
Pair of Network Cards (Or one Network Card and one Modem if you're still using a Dial-up ISP)
Cheap Video Card (It's not like you'll ACTUALLY be playing a game from this thing)
Big Hard Drive(s) and/or RAID Controller for Hard Drive(s) (the bigger and faster the setup, the better, but stay within your budget)
Floppy Drive
CD-ROM (Needed to install OS from)
To stay under $750, you might want to consider the following components:

CPU: AMD Duron; Dual Intel Celeron; Dual Pentium Pro
RAM: 128-256MB of SDRAM RAM (More memory the better
Motherboard: ATX Form motherboard with VIA KT133; 440BX; or 440FX Chipset, depending on CPU chossen
Case: Anything that fits your needs. I like them big and roomy, but those are more expensive
NICs: Generic 100/TX Network Card(s) (Full Duplex Prefered)
Video: 2MB ATI Mach64V PCI Video Card
Storage: 10GB IDE Hard Drive(s)
Floppy: Generic Floppy Drive
CD-ROM: 12x - 24x CD-ROM
Storage Controller: One can use motherboard's built in controller just fine, but a Promise FastTrak (which can me made from a $20 Promise Ultra by simple modifications) is recomeneded for fast RAID 0 or safe RAID 1
You can follow the instructions on Build your Own PC Tutorial to assemble the server hardware. When you get to the Install the OS portion of the tutorial, come back here! Then I'll talk about what OS you'll need to install to do it the cheapest and easiest way.