Yes, you can.
The TC will give you an error that the hard disk has failed. The error can be ignored but I am not sure if you can ever get the TC to show a green LED again.. ie it could flash amber till the end of its days.
I will need to check that one.
You can replace the hard disk with a small laptop one.. as long as it is sata any size will work. Mounting in the TC might be a farmer type job.. cable ties and rubber blocks.
Let me also suggest the AC version TC is worth a fair bit second hand.. cf the AC version airport extreme. You can buy a much larger hard disk for whatever usage you had in mind. And simply get the AE second hand.. !!
Opening and removing the hard disk from the TC is not exactly difficult.. but it is hard to do cleanly.. without marking the plastic.. and putting them back together is an even more difficult chore. The connectors are surface mount and will snap off the board with slightest over pressure. This device was never intended to be opened.
And the result of your exploits is one of the worst hard disks Seagate ever made.. !!