Is it possible to still back up a broken iPhone?

If the phone is toast, everything stored only on it is toast, too.

Were you using iCloud Photos on the phone before it broke? If so, the phone would have been uploading photos to iCloud. You would be able to access those photos on other devices using iCloud Photos for the same account – or through the iCloud Web interface (https://www.icloud.com/).

There may be other data that was synchronized through iCloud or through IMAP connections to e-mail accounts … where the master copies of your e-mail reside on the e-mail provider's servers.

But for anything stored only on the dead phone, it's too late.

If you can't operate the phone, it's too late to try to back it up. Backups HAVE to happen before catastrophe strikes.