Yes, it's possible but you'll be paying a lot for a very high speed connection at home but people were running home servers from home even back in the days of dial-up only. I've been running a home web server from home for nearly 10 years using just a DSL connection.
It depends what you call "high traffic". A P4 PC with 4Gb RAM, which is what I use, can easily cope with around 40,000 visitors a month. If you're talking about large files or videos then you need something a bit beefier. If you're looking at C10k (10,000 simultaneaous clients) then you're looking at specialized hardware and sofware which will be expensive.
It's cheaper and easier to use a hosting company, for web sites on a home server you're responsible for everything to do with it.
You will probably be breaking your ISPs rules if you create a home server of any kind unless you have a business account. However, so long as you don't go berserk with the bandwidth usage or do anything dumb then you should be OK. I've had no problem with mine at all in 10 years.
If your ISP has a data cap then your traffic counts towards it.
The computer you're using needs to be on 24/7 for extended periods of time - the longest I had one running was 4 years. Was it ever dirty when I finally opened it up - http://brisray.com/test/pccobweb.jpg
If there's a power failure your website goes down with it. This is why my backup power supply is more expensive than the computers it runs - http://brisray.com/test/ups02.jpg - those deep cycle batteries cost over $100 each but I can run my website for 16 hours when the mains electricity fails. The longest I've ever had to run off the batteries is 7 hours.
The batteries will last somewhere between 4 and 6 years before they need replacing. My last lot lasted 7, but that apparently is an exception. You need deep cycle batteries as they are designed to be run down to nearly flat and recharged. Car or even truck batteries will work but you have to replace them more often as they are designed to be constantly recharged. Since that photo was taken I now just use two deep cycle batteries which cost $100 each and last longer than the four old ones I used - http://brisray.com/test/upsjan12.jpg For the UPS statistics see http://brisray.com/cgi-bin/multimon.cgi and http://brisray.com/cgi-bin/upsstats.cgi?host=127.0.0.1&temp=C
You are personally responsible for everything on that server. If you get caught serving copyright material, and if you do you probably will - that server is directly traceable to you, you could go to jail and you could spend the rest of your life paying off the fines.
People WILL try to hack their way out of the web server software. See http://brisray.com/comp/cwserver2.htm , about 2/3rds of the way down the page, for some of the attempts on my server. If you're aware of what's happening and how to counter those attacks, the rest of your network, let alone the server itself will be perfectly safe. If you're running a web server there is no way of hiding that address, people need it to see your website - that's how DNS works.
Talking of DNS, you can get a static IP address from your ISP. They wil lcharge you extra for this and may even make you get a businness account. If you don't want to do that then you can use a Dynamic DNS service.