I am trying to figure out what this thing is.
I assumed it's a bell ring transformer but then it has four terminals on it.
Any help is appreciated!
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That's telephone. It's not a transformer. It's a terminal block.
Call up the phone company and ask them to have someone come out and show you where your point of demarcation is. That's where the wires stop being their responsibility and start being your responsibility (unless you have an in-house service plan as part of your bill). It looks like an obsolete one of those. If so, you can't mess with "their" half of it (the side going off to the pole).
If it is downstream of your point of demarcation, you can do as you please, at peril of breaking your landline phone service. (and Internet if it comes via phone line).
It looks like it is for telephone wires. The green grounding screw/wire at the bottom means it's probably for a outside utility service entrance like phone and not a doorbell.
I don't think you can replace it, it might not be your property even though it's mounted on your building. Verify it's phone and then ask the phone company to do something with it?