It is possible but not really practical to put Bitcoin Core on a flash drive.
Why is it not really practical? Could i put a wallet like Electrum on there?
I may have misunderstood your first question. Did you mean put Bitcoin Core on a flash drive and leave the blockchain on your computer or put both on the flash drive? I answered the latter question.
If you meant the first option you can just extract the .tgz file to a flash drive like viriat0 said.