Here's what you need to do:
Get DVD Decrypter, or any other ripper you can find.
Rip the DVD, preferrably only the scence you need.
Get ffmpeg, which is free just like DVD Decrypter.
ffmpeg is a command-line tool, so you'll need the documentation.
Use ffmpeg to cut the file to the length you want, and compress it to a format playable on the target machine. Your simplest good bet is to format it for VCD, which will be MPEG-1 video and MPEG-1 layer 2 audio.
This command should look something like this:
ffmpeg -ss (starttime) -t (length) -i (infilename) -target vcd (outfilename)
The format for times should be hh:mm:ss.xxx.
EDIT:
Don't be mad at Zero, I think he's assuming you mean you can't install any software on any machine to get this done. I'm assuming that you have some machine you can prepare the clip on, and it's the target machine that can't have installations done on it.
EDIT:
Make sure that when you set the start time and length, the start time is relative to the start of the VOB file, not the whole movie. You may need something like VLC to view the VOB if you don't have a DVD player installed on the system.