There's tons of video sharing sites. Vimeo, Flickr, Vevo, Dailymotion, Veoh, etc A quick google search will turn up lots of different sites, free and paid, with varying levels of privacy controls, even branding.
If you don't want to use a google service (youtube), go with Flickr for ease of use, privacy controls and sharing, also that 1 TB of free storage.
Such services also tend to reduce bandwidth since they will adjust resolution for streaming which may be an issue if you've got an entire office suddenly deciding to watch the same video at the same time in individual offices. Like Toby said, you probably won't be able to build a better infrastructure than a service that specializes in it.
