It is easy to download the audio of a YouTube video using youtube-dl. It's also simple to extract only a section of the audio using ffmpeg. For example, on a Mac, the following command will download the audio from E^st's cover of Bitter Sweet Symphony, starting at 45 seconds into the clip, and ending at 5 minutes and 13 seconds:
youtube-dl -f 140 https://www.youtube.com/watch?v=s64RnXSwn-A -q -o - | ffmpeg -i - -ss 00:00:45 -to 00:05:13 'E^st - Bitter Sweet Symphony.m4a'
-f switch specifies the download format (where 140 requests 128K ACC in an M4A container),
-q sets "quiet mode", and
-o - is used to send the output of youtube-dl to STDOUT. This output is piped into ffmpeg, where
-i - reads from STDIN,
-ss sets the start-time offset, and
-to sets the end-time offset.