You can use a cheaper hosting solution to host your video website, but the videos themselves will need to be streamed from a dedicated (and more expensive) service.
If you really want to roll your own, you'll need at least a good server to accept video uploads and transcode them to the different formats (there are probably at least 12 you'll need to worry about right now, and likely more). You'll also be paying for bandwidth in and out of this server, so budget for that. Streaming video is best delivered from a CDN built for the purpose, and that can be expensive (again, mainly bandwidth). You'll need a reliable, scalable storage solution (like S3) to store all 12 copies of each video.
You will need some sort of oversight to prevent copyrighted and illegal works from being made public. If you're delivering h.264, MP4, AAC, or MP3 streams, you'll have some licensing costs to pay to MPEGLA.
Unless you're planning on just hosting curated YouTube and Vimeo links, you won't be able to do this on the cheap.
Relevant experience: I've built this 3 times over the past decade for 3 different companies; it's never cheap, it's not quick, and there is always someone you forgot to pay a license fee to who will send you lawyer letters.