The event occurred is 26,000 light years away, we date it as having occurred 26,000 years ago.
How is it that a location that's closer, say 10,000 light years, would have dated the event to 26,000 ago rather than 10,000 light years?
ds^2 = dx^2 + dy^2 + dz^2.
In Special Relativity, you define a quantity called the spacetime interval:
ds^2 = -(c*dt)^2 + dx^2 + dy^2 + dz^2.
You treat time as a 4th dimension, and take it into account when calculating distances. Notice that this interval can be negative (the metric is "semi-Riemannian"). "Events" (locations in 4D spacetime) can only be causally connected (one can influence the other) if the spacetime interval separating them is negative (this is equivalent to saying that no information can ever propagate faster than light).
The key is that all non-accelerating observers agree on the spacetime interval between any two events. The spacetime interval between two events is the only solid distance you can give. The time interval by itself depends on the observer's reference frame (you've probably heard of time dilation). So too does the spatial distance (length contraction). But time dilation and length contraction behave in such a way that the spacetime interval remains constant.
The long and the short of it is that time intervals are observer-dependent.
I can't imagine that they would say that!
I'm sure that you mean the equivalent of 24,000 BC in our reference frame, after Lorentz transformation. Not that they would likely even know about us, or manage the transformation.
This is bloody confusing.
Or know when ours was supposedly born?
This can happen for instance with extreme differences in relative velocities. Someone else can probably show an example with the math, but a key concept here is the that distances also change when you move real fast or are in a deep gravity well.