Design+programming front-end webpages: $150,000.00 for a start.
Backend (server architecture, converting the videos etc.): at least the same as above.
In other words: Now, that you already know exactly what you want, you could set up a fake youtube for roughly a quarter of a million. Anything below might look like youtube or "simulate" youtube, but by no means be exactly like youtube.
A quarter of a million dollars? Man, I'd love to work for any company that overpays so enormously.
Heck, the company I work for just got a contract to build an eBay style web site for a client. Since I'm the only one on staff with programming experience, it fell to me to do the entire project...total cost so far (including my salary and a few minor 3rd party software packages) is under $15,000 USD. We've got maybe another week of development to finalize a few features, but we've already done testing of everything that's in place so far...even if the final cost ratched up to $20,000 USD, I'd be amazed. We've included all the features that eBay has, and have ended up with something almost identical to eBay itself (except, of course, for the fact that it's intranet only, and doesn't have a big shiny eBay logo on it).
Basically, if you have any programming experience (I'm an electronics engineer, but took several years of computer science courses at university, as well as programming in my off time for the last decade and a half), you can slap together a reasonably complex website in very little time, and for relatively low cost. Estimates of $150,000+ for design and programming only like the one above are what are normally called NPFSA (Numbers Pulled From Someone's Anus) estimates.
Of course, all this depends on the traffic you expect the site to be able to handle. The site I've built has been scaled for a corporation's internal traffic, with 65,000 total users, and an upper-end expectation of 1,000 users simultaneously. If you want something that complex that can scale to millions of users, then you'll need to spend a great deal more money on hardware and bandwidth.
And then there are advertising costs...ongoing maintenance and administration...accounting...
There are plenty of costs involved with a large-scale site that go way beyond just programming and hardware, and those usually far outweigh the actual cost of developing the site itself.