Play video inside the Unity from External Cloud Storage?

Which external cloud service provider would you use? It would depend greatly on which one you pick. Assuming they have an external API of some kind that allows you to link or download the video, then yes, you can do this. The details depend on where the video is completely dependent on where the video is coming from.

Since all of the .net framework is available, you can write a plugin to load or stream form any source you like.

There are also a few plugins already available on the asset store. Here is one that works with YouTube: https://www.assetstore.unity3d.com/en/#!/content/19138

If you decide to write your own, I would recommend the red5 media server, because there are already many .Net libraries you can reference to stream the videos, and it would be from your own server, and with your own ads if you chose to have ads. You can also stream from camera or any other source you like: http://www.red5.org/

But you will need to integrate it into unity yourself.