It is possible to host your own website from home, as long as you can fullfill the following requirements.
If you plan on running a mail server out of your home, you will also need port 25 to be open, which usually requires business class Internet Service. Most ISPs will block port 25 for residential customers, they do this to help reduce compromised systems from acting as spam servers.
A dedicated IP address is also recommended. Most Business Class connections come with 1 or more dedicated IPs. On the residential side. Cable Modem connections typically have very long lease times on IP addresses, for that reason you might have the same IP address for 18 months if not longer. DSL connections often rotate IPs every few days, making them a bad option for hosting your own website, as every time the IP changes you have to update your domain's DNS records to reflect the new IP.
Speed is another factor. Most business class internet connections will give you plenty of Upstream, but residentual connections tend to give you little upstream and lots of downstream. Upstream is what you need to be able to host a website. DataCenters often give customers Gigabit ports (125MB/sec). While your home internet connection might only give you 4 Mbits (500 KB/sec ).
Even though you could host your website from home, its generally not recommended for performance reasons.