It can be done, but most connections (uplink) are too slow for moderate traffic sites. Also, running e-mail is a bit of a hassle, and your IP could change at any time (meaning downtime, usually a few hours, every time it does).
Personally, I would not even bother with it. You are much better off purchasing a small package from a hosting company.
heh, i forgot about ARIN here in Europe we have more freedom with ip's. But, for dynamic ip i can recommend no-ip.com. They offer good solution to them who want use domain or subdomain on dynamic ip's.
Rustelekom LLC Dedicated server since 2002, RIPE NCC member, LIR, AS51168
You can always point mydomain.tld -> CNAME mydomain.dyndns.org -> IP
but the others are right, the changing dynamic IP will screw up your email.
Especially since it will be more difficult to e-mail anyone using spam filtering and RBLs. Spam filtering systems are much harder on e-mails when they originate from a server using a dynamic IP address.
There is also the little fact that most ISP's prohibit running a dedicated server through your home internet connection. You can probably get away with it if you want to just play around to see how things work, but for long term, constant traffic, you'll probably end up getting a letter or phone call from your ISP at some point.
I know some people who do it, but they have business accounts. And, as others have stated, for that cost, you can buy dependability, security and battery-backup in a data center for less money.