Simple file sharing and backups at home

Does your ISP issue a fixed or dynamic IP address? My ISP issues a dynamic IP address so my IP is issued to me randomly. I use DynDns to fix that (its free).

As for ipv4 addresses, just go to System Preferences>Network>Airport>TCP/IP and under configure ipv4, choose DHCP with manual address and proceed to key in the IP you want. Like 10.0.1.1 for example.

If you're looking just for networked file sharing/backups at home, not visible from afar, then an Airport-plus-some-drives or a Time Capsule, or a Mac Mini (or any of a wide variety of other home "NAS" -- Network Attached Storage -- products) would work just fine. I'm partial to the (industrial strength, but spendy) QNAP and ReadyNAS boxes myself, though I don't own either yet. Note that everyone and their brother are hopping into the NAS market these days, including all the drive makers and accessory makers, and while it isn't rocket science, it is easy to mess up quite spectacularly -- if you're looking to buy a NAS from a company that doesn't have a long history of specializing in such things, check carefully for trends of negative reviews before buying.

If you're looking to be able to access this server from offsite (i.e. connecting from the office, or a friend's house), then you should have a very careful look at your ISP's Terms of Service (or Acceptable Use Policy), as many of them forbid running an (externally visible) server of any kind. Many people do anyway, without getting in trouble; I'm just pointing out the possibility for being on the losing end of an argument with your ISP at some point.

As far as IP's go, if it's on your private home network, no problem, it should get a private (10.#.#.#, 172.#.#.#, or 192.168.#.#) address from the DHCP server on your existing WiFi router automatically. If you want to be able to access it from outside, you can either teach the WiFi router to port-forward whatever ports are needed from your external (cable/DSL/etc) IP to the private IP of your Mini (or whatever), or you can see what your ISP offers (some plans may allow you to have multiple external IP addresses), though if they forbid servers, this may lead to an awkward conversation.