How to stream my video files from the cloud? I use Dropbox to backup all my important files, but I'm running out of storage space on my computer.

Many of the files are video files that take up the majority of my storage, and it's not imperative that they're stored on my computer.

Is there a way to configure Dropbox so some selected files (in my case the videos) are stored in the cloud so I can stream the videos when I need to watch them, but otherwise they don't take up space on my local drive?

Or is there an alternative cloud service that provides this functionality?
You could try Jungle Disk. It's a pay service, but it gets you what you want - a locally mountable cloud drive. It can store data locally as well, like Dropbox, but you can configure it as an archive, with no local storage used. It uses Amazon S3 or Rackspace as the backend storage, so you pay a $3/month fee, plus an amount per GB per month.

I believe there are similar services out there, and you can just use S3 directly and get something like Panic's Transmit to mount the S3 storage as a local volume.