Simple Extrude?

This is not a question about 3D modeling software.

Say I'm interested in constructing a simple bookshelf. Everything is to be made from 3/4" plywood. Can I draw a face of the plank and extrude it so that it is 3/4" thick? Then construct other planks, extrude them, and put the pieces together into a bookshelf (then dimension it, etc.)?

No, LibreCAD is strictly 2D.  The best you can do is an isometric view.  Refer to the User Manual for a few details on isometric drawing.

JR wrote

This is not a question about 3D modeling software.

Yes, assuming you mean cut rather than extrude. You will get a drawing file and how you use it will depend on your CNC software.