How do I backup my SSD when my MacBook doesn't even turn on? I'm
guessing I need to remove the SSD but I don't know what to do after
You are correct; you just need to remove it.
What you do afterwards is put it in a safe place (like a desk drawer).
There's no need to erase, much less even have, an internal SSD to diagnose and repair a power issue.
If the "Apple Center" (you didn't say if they were an actual Apple Store or an authorized 3rd party repair facility) said that "wiping" the drive was necessary, I would look for another facility.
You can boot a MacBook from USB, another SSD, or from another MacBook (via Target Disk Mode) so there's no reason that they should get yours (as a side note, they should have spare ones with clean OSes for diagnostics). An internal drive is absolutely not necessary for diagnosis/repairs (unless of course the issue is the SSD itself). As it's a power issue, the SSD is irrelevant and shouldn't even be touched.
The problem that you are experiencing is related to the DC-in board (aka the MagSafe board).
Besides being the interface for your charger to your MacBook, it acts as a safeguard to ensure that you're not dumping 18V DC into your SMC (which handles power) that only takes 3 volts. (It's also very inexpensive).
It's not a complex fix, but it is detailed as you do have to remove the logic board. iFixit.com has an excellent step-by-step walk through on how to do this. However, if this is not your cup-of-tea, I advise taking it in.
That said...you should be backing up on a regular basis.