I would like to have a 64bit Linux virtual machine running just like another terminal window in my 32bit Linux Vx-enabled laptop. How can I do that? I don't need a GUI for the virtual machine, just a bog-standard terminal-based Linux VM.
You'd need a VM host that will let you run a 64 bit os on a 32 bit system - virtualbox will do fine with a VT enabled system. Do a basic installation including a ssh server. Run it in Headless mode after that. Then you can open up a terminal window and ssh into your guest.
Your instructions may vary based on the VM software you choose to use, but for most part this should work fine.