How do you dramatically reduce the size of a video while keeping the quality as much as possible?
I'm not familiar with handbrake, but this is basically a math problem.

First, you have to decide on a codec. One of the 'best' codecs in regards to quality/filesize-ratio as well as device-compatibility is H.264, so I would recommed you go with that.

I've done a quick search on handbrake and it appears you can set the quality in two ways, either by choosing a Rate Factor (RF) or by setting an Average Bitrate (see screenshot, correct me if I'm wrong).



Source: Wikipedia

Since you want a ratio of 700 MB per 1 hour of video, let's do the math:

700 MB = 700 x 8 Mb =  5600 Mb
5600 Mbit / 3600 s = 1,56 Mb/s = 1560 kbps


So to achieve this filesize average, set your Average Bitrate to 1560 kbps. This is quite low for HD video, so make sure to check the 2-pass encoding option, which will increase the quality at the cost of a longer encoding time (it does so by doing a first check or pass on the video as to how high a bitrate is needed for every part of it, so the encoder can better adjust the bitrate accordingly over the course of the video).

One final note, the video you linked recommends you change the audio bitrate to 128 kbps. This is a bad idea. Given that we are already using up 1,56 Mbit per second for the video, the few additional bits per second that are needed for the audio won't make much of a difference regarding the resulting file size. And 128 kbps will sound bad; we are much better at detecting flaws in audio than in video. I highly recommend you set the audio bitrate to at least 256 kbps.