The first issue that you’ll have to worry about when running a server from home is that most home internet plans specifically have a clause against running a server. So if the automated network monitors at your ISP start noticing a lot of incoming connections to port 80 you may get a nastygram or just get cut off until you talk with your ISP.
The next issue is that the internet is an evil, vile place full of griefers. So don’t be surprised when you find you have zero bandwidth or have hit some random transfer quota because some asshole decided to download your videos until your ISP shut down your connection.
Third, because you’re going to be serving videos you’ll want to make absolutely sure you have the right to distribute those videos. This means you either have to make the videos yourself or get it in writing from the creator with some verifiable proof that it’s your video. (I’m not sure of how you would prove it’s your own video.) Be prepared for some DMCA griefing whether by jackasses or genuinely mistaken individuals.
After that, the technical issues of maintaining a server aren’t so bad. Dell makes $700 rackmount or $600 desktop servers that should work until you have a sizeable user base (at which point you’ll want to go to a hosting company for reliability sake.) Keep server backups and remember that RAID is not a backup tool. Use a LAMP stack (the L stands for Linux) or use a BSD as your OS. (I think it’s OpenBSD or FreeBSD that has 24 years of semiannual updates with zero security bugs.) Ubuntu is easy peasy to start. There’s plenty of free blog software out there – just make sure to read and comply with the license of whatever you use. (Most free blog software has some clause in the license about debranding it – and for good reason.)
If you really have to cheap out you can sometimes find a newer-but-not-newest server on eBay for $400-$500 shipped. Or, check out ABMX. They have <$500 1U rackmount and tower servers.
http://www.abmx.com/entry-level-tower-server should be a decent starting point. Once you get a large enough userbase you want to consider a hosted solution, possibly using Amazon S3 for distribution. You might want to consider distribution through torrents as well and having your server be a seed box and tracker as well.
This may have been somewhat rambling but I think there’s enough information there to start on. Good luck on your new endeavor!