Linux detects an external drive as absurdly large

Based on your smartctl output, the problem isn't that the physical storage of the drive has failed; instead, the fact that it can't even correctly report the size (which should be "burned in" to the firmware) suggests that the controller electronics at some point in the chain have failed.

Most external USB hard disks have basic but not always advanced SMART support; Seagate drives have been known not to provide even the baseline expected support, so I'm not willing to say for certain that the SMART failures are a result of the controller failure. However, the inaccurately reported size indicates that's what happened.

If this is an "all-in-one" external drive, you might be able to recover the data by removing the actual hard disk and connecting it to a standalone USB-HD adapter. (This would be the case if the failure is in the USB interface electronics and not the onboard drive electronics.) If that doesn't work and you get similar errors, your data may be recoverable if you send it to a recovery lab, although this tends to be very expensive and only worthwhile if you have something like priceless family photos or a valuable Bitcoin wallet.

I'm afraid your drive is gone. The error clearly suggests that it has bad sectors and it tried to reallocate them but clearly there are no viable sectors left.

Looks like the HDD has been accumulating bad sectors for some time. It eventually has no space available for any more reallocation.

You can try running smartctl -a /dev/sdc to see if you can get additional information.

Edit

The smart data indicates that a problem in firmware is causing this issue. I've used drives that had more than a thousand bad sectors. Yours just seem to occupy the blocks required in initialization.

You may try and find out if Seagate offers any software solution from this link

Also, professional data rescuers might be able to fix the drive and the data. However, they'll probably cost way too much. So I don’t support going that way unless there’s something invaluable in the drive.

Additionally, you may try some hardware solutions on your own. Here's a simple tutorial from Youtube. However, you may lose both the drive and data if not careful.

Debian 5.5? That's a really old version so you may have hit upon a bug in lsblk where it's showing some max value for a dead drive. The drive does appear to be dead from the log you've attached.