I am trying to find out where can I get a shared video hosting (if
there is a thing) like webhosting. However, I wanted it for videos. I
currently stream live. But would like to have my own server that I can
push to

You should check out www.micfo.com, for the price they get you an
incredible amount of Bandwidth.  I believe you can get 1000gb a month
of traffic for as little as 150 dollars a YEAR.  Its shared hosting,
but they do give you SSH and FTP access as well as CPANEL (A user
manager to manage all aspects of your web host).  Hope this helps.

Lets not miss the obvious...
https://upload.video.google.com/

Try http://youtube.com/