Hosting it yourself is almost always a bad idea. The minimum cost of doing so is hundreds of dollars a month, if you want a reliable site that stays up 24 hours, 7 days a week, and provides professional quality speed to your visitors. You can't do it on a cable modem -- most cable ISPs forbid this and will shut servers down. Cable modem upload speed is usually 128kbps, which means that you cannot serve content to others anywhere near as fast as you can download content yourself. DSL providers are more friendly to servers, but they also typically have limited upload speeds, and in many cases will not guarantee high-quality performance for servers run over DSL. The gold-plated option, a "T1" line, costs $600/month from the most affordable providers such as Speakeasy.Net.
Compare the price of any good, reputable hosting company as found on webhostingratings.com. For less than the money you will spend to host it yourself, you can have your own independently rebootable box with full scripting and Unix shell privileges, and multiple redundant Internet connections much faster than a T1 line. For more information, see the "how do I set up a web site?" entry.

There are only three situations where do-it-yourself hosting makes sense:

Hosting companies will not tolerate your content. This is a rare situation. Even adult oriented sites that are not welcomed by some mainstream hosting companies have a choice of hosting providers that will gladly work with them.
You are willing to pay the price for an extremely fast, professional grade connection at home or in your office, you would like to offset that price by using the connection to host your web site, and you don't mind being your own system administrator.
You really, really, really want to.
Those who "really, really, really want" to host their own sites should pick up a good book on the subject, such as Open Source Web Development with LAMP. Note: I wrote a forward to this book but do not receive any royalties. I genuinely recommend it and learned a lot from it myself. As for your connection, consider Speakeasy business class DSL or a T1 connection. Of course there are other quality providers; I get my own T1 connection from Sprint, but their prices are much higher.