Trying to watch a video on Ant1's web site displays a black square saying:
"Error loading player: No playable sources found".
An easy way is to use Livestreamer.
Install Livestreamer (docs)
Update: Livestreamer hasn't received updates since Feb 2016 and looks abandoned. Streamlink is a fork that looks active.
Make a note of the pages' URL, for example:
$ livestreamer 'http://www.antenna.gr/webtv/watch?cid=XXXXXXXXXXXXXXX' best
it should open the stream, automatically detect your media player and start playing the video.
Save video stream
In case you have problems watching the video, you can add the
-o <file> parameter to save the stream to the disk:
$ livestreamer 'http://www.antenna.gr/webtv/watch?cid=XXXXXXXXXXXXXXX' best -o video.mp4
This workaround is here for educational purposes, as the method is relevant to streaming in general. Individual sites may have their own terms and conditions that may or may not allow using their content this way. It is your own responsibility to read, understand and follow these rules.