There’s more to the conversation. HandBrake does not access your system's codec component configuration so many codecs supported by QT on your system may not be compatible with HandBrake while formats not supported by QT may be supported. (I.e., HandBrake supports FFmpeg compatible compression formats.) Quicktime reads and plays all my.mov files.
iSkySoft makes something called iMediaConverter for only $49. I use it all the time, and it does exactly what you want it to do. You pick your source (in your case the DVD disk), and then what you want the output to be. In your case you would just choose the preset for the iPod Classic. It works really well, and is really easy. Note though that it rips your movies into one continuous video file, with no chapter markers. Not sure if this is even possible in any video file playable on an iPod.
In any case, I feel like iMediaConverter is a steal for 49 bucks. It basically converts any video to any other video format you want. I've used it to convert .avi files that came from a Windows computer as well.
HandBrake is a free, downloadable program that allows users to convert video from nearly any format. Unfortunately, HandBrake does not allow audio-only conversions, that is to say, the answer is “no” to the question of “Will HandBrake extract audio?” Hence, a good HandBrake alternative will be your better choice to extract audio and this easy-to-use as well as useful audio extractor can be free downloaded here:
https://www.videoconverterfactory.com/download/dvd-video-converter.exedvd-video-converter.exe