I have found a way to get rid of this at the sacrifice of some of your gear:

1. Start by dropping your backpack somewhere hidden that you know (make note of the server first so you can come back later and get it). It will not show up on the ground, you will have to quit and re-join the same server to 'refresh' your inventory.
2. Leave the area, run at least 1000ft away.
3. Exit and re-join the server. If the issue is fixed now, it was caused by an item in your backpack. You can now return to your backpack, pick it up and try to just drop one item at a time.
4. Continue dropping items and leaving the area until you either found the item causing the issue, or have enough of your items to be happy.

This issue for me was caused by either a fire extinguisher in my backpack or a stack of wood (for fireplace).

Same issue from Central US on nearly all servers. Cannot put anything in hands, interact with anything, vault or consume items. Can walk around and position updates on different servers.

Tried dropping my backpack, it appeared to not work, but when I joined another server, my backpack was gone. Rejoined the previous server and sure enough, my backpack was on the ground there. Tried picking it up, it appeared to not work, but when I joined another server, my backpack was picked up.

To me it seems like an issue with server -> client updating. The actions may be happening, but the server does not convey this to the client.