Here are some considerations for running a home-server that hosts a small site (max 50 simultaneous connections) running *NIX. These are what I consider to be the minimum required.

1. Hardware
Minimum hardware requirements for CPU and RAM would be a Pentium 2 350-450MHZ with 512M RAM. The more services are running, the more ram you need.

2. Connection
A minimum upload speed of 80k/s is recommended, to be able to serve your site at the minimum acceptable speed. Your ISp must allow you to run servers and not block or throttle important ports like port 80.

3. DNS
A static IP is useful but not necessary in all cases. Use free DNS servers or the one from your domain registrar. In case of dynamic IP use a free dynamic DNS server. If you plan to run your own public DNS server, you will need 2 IP's with at least 1 static for secondary DNS server.

4. Software
From operating system choice to applications choice, you will need to configure, optimize, secure and maintain a lot of software. While the software is free, it takes some time to learn it. It can be fun and frustrating, but worth the trouble once you master it.

5. Other
The equipment will be running 24/7. It will increase your electric bill, produce heat and noise. Dust will eat up fans fast. Placing the server in a remote room, possibly closed clean room with no heating, is important if you plan to live with computers and not go crazy, or have pets or parties at your place.

This is all I can think of right now