For the sake of abilities, yes it's possible . You do not even have to have static IP, you can setup dynamic DNS update with many available services on the interwebs, many of them are free of charge, but from the economic point of view it's not good in most cases. Running the server at home might be problematic, loading of pages will take much longer than on professional hosting + you would have to invest in UPS so that your server is up and running when the power is not supplied.
If you want to train your skills it's very good idea, I myself have been playing with debian + apache + mysql , ssh and other things long time ago and have setup the server at home just for fun, but if you want to treat the IM stuff seriously it's a lot better idea to invest in either cheap vps or shared host. You will save lot's of time since almost all hosts these days have easy to use panel with options to install lots of useful scripts like wordpress or drupal with a click of a button.
