What's with this weird kitchen 120V circuit with 240V on it?
It's a Multi-Wire Branch Circuit (MWBC)
Which is a thing that is new to you, and you'll probably need to do some learning about it.  To cut to the chase: To hook up a GFCI outlet here,

note the warning tape on the GFCI that says &quot;Do Not Use. For Wizards Only.&quot;  Do not remove that tape. In fact it will be important that you follow that rule anywhere on this or any MWBC.
Most GFCI receps have a mechanism for accepting 2 wires on the LINE screws.  Both black wires go on LINE brass.  The white pigtail (after you put all that back together) goes on the LINE silver. Don't even think about using the &quot;2 wires under a screw&quot; feature to attach the 2 white wires and skip the pigtail. That's a safety issue!
Ground wire of course.
Button it up and you're done.

More on this &quot;Multi-wire branch circuit&quot;
A MWBC is a setup with 2 hots wires sharing a neutral.  The neutral carries only differential current, which can never be larger than the max current on a hot wire.  Thus the circuit breaker is able to protect the neutral.   Certain rules must be followed for safety:

The neutral wire, where shared, must be pigtailed. That way a device can be removed without severing the neutral wire for the other half of the circuit.  As you saw, when you do that, things got wackadoodle.
The two breakers MUST be phased opposite, so there is 240V between them.  That means breakers must be very carefully placed, especially if double-stuff breakers are involved.
The two breakers need to be handle-tied so that turning them off together is compulsory.  You can buy special handle-ties for those breakers, but a far better plan, especially for a novice, is to use a 2-pole breaker.  This has a factory built-in handle-tie, and better yet, it mechanically enforces the correct opposite phasing.
Because neutral is shared, the LOAD terminals can never be used on GFCI receps.  (and novices should not be using them anyway unless they know exactly what they do, and I don't mean &quot;give me a place to land these other 2 wires&quot; :)

You know how many GFCI receps you have to install, and how much they cost.  If you use a GFCI 2-pole breaker, ka-zam! you GFCI protect the entire circuit (both sides) in one step. Not only is this safe, it can also be more economical if you have 4+ receps to replace, depending on the obsolescence of your panel.
Those measurements
The measurements you took were 100% consistent with a MWBC in which an appliance on the other circuit had been switched on.  The red wire is 240V away from the black; imagine there's a resistor between red and &quot;Hot White&quot; (really downline white).  With zero current flow (since you interrupted it), the &quot;hot white&quot; is being pulled up to the red wire's voltage.
I suspect this circuit also has a 240V load somewhere on it; that would act like a resistor connecting red and black.  That explains why supply-side black was 240V away from downline black.   Given that a 240V load is present, you cannot use a handle-tied breaker, and must use a 2-pole breaker. Although a GFCI 2-pole will be just fine.