Why does spatial distance measurement work differently than proper time measurement in general relativity? In §84 of Landau and Lifshitz's Classical Theory of Fields, L&L talk about measuring distances and time intervals in general relativity.
They begin by finding the dependence of proper time $\tau$ on the coordinate $x^{0}$. For this, they consider two infinitesimally (time) separated events, occurring at one and the same point in space. Jumping ahead, setting $dx_1=dx_2=dx_3=0$,
$$
ds^2 = c^2d\tau^2 = g_{00}(dx^0)^2. 
$$
Next, to determining the spatial distance $dl$, they do not set $dx^0=0$ in $ds$. They claim that one cannot do this since, in a gravitational field, proper time at different points in space has a different dependence on the coordinate $x^0$.
The above explanation is not sufficiently clear to me. I get that length measurements can get tricky in relativity. For instance, while discussing length contraction in special relativity, simply making a Lorentz transformation to a relatively moving frame and checking how the distance between the head and tail of an object changed does not work, since one has to make sure that the measurements are made at the same time.
In summary, I do not see the problem with simply putting $dx_0=0$ here. Why exactly does it not work?
 A: It's hard to see what L&L are aiming for in this section since they don't seem to define the "spatial distance" whose properties they're trying to derive, but working backward from the conclusion, I think I understand.
At the beginning of the section they say

We have already said that in the general theory of relativity the choice of a coordinate system is not limited in any way; the triplet of space coordinates $x^1, x^2, x^3$, can be any quantities defining the position of bodies in space, and the time coordinate $x^0$ can be defined by an arbitrarily running clock.

So in spite of the first half of the sentence, they aren't considering arbitrary coordinate systems, but only those where the $dx^0$ axis is timelike and the other axes are spacelike; and they want to interpret this physically as a network of clocks with $(x^1,x^2,x^3)$ labels, like those traditionally used in special relativity.
$dl$ represents the proper distance between two nearby clocks, the ones with the labels $(x^1,x^2,x^3)$ and $(x^1+dx^1,\ldots)$. It's a distance between worldlines, not between events. You can't calculate it by setting $dx^0=0$ because if the coordinate axes are not orthogonal then you will end up measuring the distance diagonally, and get a smaller (Lorentz contracted) result.
This is not a standard approach. Classical field theory on a GR background is normally done in a manifestly covariant way, at least these days.
