Sure, you could do it with .htaccess, the problem is that you only have a couple of options:
1. Allow any "key" matching a specific pattern to access the content
2. Store all of the "keys" in your .htaccess and iterate through them every time a URL is accessed
The Vimeo example you posted is using a unique session Id to track a user around the web site. That user's credentials are tied to that session Id so when that user tries to access a video, all the server needs to do is cross-reference the session Id with the user's permissions to see if they have access to the requested URL or not. This is common practice among larger sites.
I promise you that Vimeo is using cookies and scripting to manage that level of access to their assets. .htaccess is bad ass, just not for that kind of work
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If you know PHP, you can code something up like that in a few lines...let me know if you need a starting point.