Streaming video in your home can be a complex problem but with the right combination of software it could be achieved quite simply. In order to accomplish this we first want to make sure all our videos are in a compatible format. Most videos currently stored on the backups can be converted to an appropriate format using the following command on Ubuntu.
avconv -i input.mkv -c:v copy -c:a aac -strict experimental -b:a 192k output.mp4
Combining this with technology such as VideoJS and your solution is close to completed. Of course these needs some sort of http server but using Apache or something similar this may be set up in a jiffy. Only thing really missing is a password protected administration system which may require some sort of LAMP stack.