Full Disclosure: I am the lead developer on the site I'm mentioning...
If you can get the URL to the .FLV file (just google 'how to get FLV URL youtube'), Upload by Transfer is what we call it. That feature gets released next week on our platform (see blog post for details). You provide the URL to the FLV, and the cloud service pulls it off and transcodes it.
Edit: Just tried this with the KeepVid suggestion in another answer. Just picked a popular YouTube video... http://play.nimbushd.com/view/lfsv
This is normally meant for production shops with automated workflows (HTTP/FTP servers) but FLV is supported and it's no different. And in the interest of neutrality I suppose you could also do this with encoding.com if you had a destination for it (an FTP server or something).
If you're not looking for a cloud service/hosted platform, then of course there are YouTube downloaders and you can just save the .FLV files off to your hard drive.
I can see that you are using FlashGot addon for Firefox. FlashGot determines the direct link to the video and makes it available for download as soon as you start viewing a YouTube video; and each time you step to a different point inside a video, FlashGot creates another link for that split part. Your best bet would be to use the first link FlashGot provides.
If you are still looking for alternatives to FlashGot, see the answers to this question.
Best Answer
If it's your video, go to Analytics > View reports > Playback locations > Embedded player on other websites: