The requirements to handle 1000 unique visitors/day will depend on what exactly you are serving. I ran a website that got well over 1000 uniques per day on a Pentium MMX 200 Mhz with 64 megs of RAM and a 1.2 gigabyte hard drive. This was significantly overkill for the site. However, that was entirely static content. Oh, except it handled email, spam filtering, and a database for a POS system for a retail establishment with two stores.
If you are serving mostly dynamic content, you'll want more processing power and more RAM. Almost certainly, you'll be fine with a bottom end computer, but you probably want something manufactured in the last five years or so. This will obviously depend on what your dynamic content actually is, though; more complexity will require more processing power.
If you cannot afford any outages, you may want to look at redundant hardware, failover systems, etc. etc., but you first need to determine how much an outage will cost you. What if you have a 5 minute outage? An outage lasting an hour? Eight hours? A day? In any case, before you look at redundant hardware, you'll need a service level agreement from your ISP.
And of course, if you are looking at something to stream 1 gigabyte of traffic to each of these thousand uniques, that's a whole different matter. Now you may want to look at content delivery networks, and possibly multiple servers just to handle the outbound network traffic.
No matter what your requirements, though, you need to look at a good backup solution.