How to back up my MAC if it can turn on because the Water damage?

Backing up is something you do regularly before bad things happen (because they will). If you can no longer turn on your computer, it's too late to back it up.

Make an appointment at the Genius Bar at your local Apple Store for further evaluation:

Genius Bar Reservation and Apple Support Options - Apple

You may be able to back it up if you can access the Mac in recovery mode, however I find this unlikely if it is water damaged.

See https://macreports.com/how-to-back-up-your-mac-that-wont-boot/

If you can't boot in recovery mode, then there isn't really any way you can back it up unless you take it into an Apple Store, they may be able to help.

Did you say what exact mac / model /year is this?

If this the 2016— Apple’s in house customer data migration tool is specifically designed for the 2016 MacBook Pro and includes a logic board holder with power adapter that allows repair staff to insert your logic board and connect it via USB-C to another MacBook Pro to salvage data off any logic board, SSD, Touch ID, or wireless card failures,

Sorry it wasn't better news.

Best of luck.