This question is about the meaning of the length measurment of events,
from a particular observer, in terms of the metric tensor of the
4-dimensional space-time.

Consider an observer. This is represented in the space-time manifold
by a time-like curve. Now consider of an event taking place at some
point on the same manifold, not necessarily on the observer's
world-line. I want to know about the measured distance of the event
in view of this single observer.

I am not sure of the answer. But there are two quantities which seem
to have some relevance. (1) If the event is not too far from the
observer, there is (I thinjk) a unique space-like geodesic connecting
the event to the world-line of the observer. The 4-dimensional length
of this geodesic, \int |ds^2|^(1/2) along it, is a candidate, althoug
I don't know exactly what this quantity means. (2) The light-cone of
the event interesects with the world-line of the observer at two
points. These points can be thought of as the events of respectively
sending and receiving a signal by the observer to and from the event.
The time elapsed between the sent signal and the received one as
measured by the observer's clock, i.e. the 4-dimensional length of
this portion of the observer's world-line, \int |ds^2|^(1/2) along it,
times c/2 is another candidate. This latter alternative very much
resembels what we do in our experiments.

Even if neither of the above is correct, what is the significance of
them? And what is the true measured distance?

Thanks for attention.