How can I do a backup if my computer won’t startup

As stated above, you cannot. You'll have to fix the startup problem.

If your Mac doesn't turn on - Apple Support

If your Mac doesn't start up all the way - Apple Support

Will the Mac startup in Safe mode (hold the shift key down at startup). It can take much longer to safe boot (10 min) so be patient.

How to use safe mode on your Mac - Apple Support 

Safe mode disables loading third-party drivers, forces the OS to clear certain caches and logs and does other miscellaneous housekeeping. It's non-destructive to user data so is safe to do. Very often certain weird software behavior is corrected by simply logging in this way as a troubleshooting method. When you're in safe boot, the machine will not be at its best performance, especially with graphics, but that's expected.

Does the problem persist while in Safe mode?

Exit safe mode by restarting your Mac normally and re-evaluate the issue again.

You cannot. you'll have to use an older backup.