How to extract only the raw contents of an ELF section?

Use the -O binary output format:

objcopy -O binary --only-section=.text foobar.elf foobar.text

Just verified with avr-objcopy and an AVR ELF image’s .text section.

Note that if, as Tim points out below, your section doesn’t have the ALLOC flag, you may have to add --set-section-flags .text=alloc to be able to extract it.