Distance in General relativity I read a few lines about general relativity and one of the first equations is the one defining the eigentime of a time - like curve. But observers should also be able to measure length, right? So is there also an equation for the "eigendistance" of a time-like curve?
 A: The proper time of a time-like curve is its length.
A: 
I read a few lines about general relativity and [... an equation for] the eigentime of a time-like curve.

I suppose that this is referring to an equation similar to
$$\tau A_J^Q := \int_0^1~dt~\sqrt{g[~\dot\gamma, \dot\gamma~]},$$
where


*

*$A$ denotes a particular participant ("material point", "principal identifiable individual"),

*the quantity being expressed, $\tau A_J^Q$, can be called the arc length of the time-like curve consisting of events in which $A$ took part from the event of $A$ and (a suitable, unique) participant $J$ having met (and immediately subsequently left) each other, to the event of $A$ and (a suitable, unique) participant $Q$ having met (immediately after having approached each other);
or more concisely (and arguably more directed towards physics) $\tau A_J^Q$ can be called $A$'s duration from having indicated $J$'s passage until having indicated $Q$'s passage,

*$g$ is called the metric tensor (of the space-time region $\mathcal S$ under consideration, as a function of two suitable time-like arguments according to the sign convention illustrated here),

*$\gamma : [0 ... 1] \rightarrow \mathcal S$ represents the (throughout) time-like curve under consideration, with

*$\gamma[~0~] = \varepsilon_{AJ}, \qquad \gamma[~1~] = \varepsilon_{AQ},$
and the image of curve $\gamma$ is without gaps and contains only events in which $A$ took part,

*$\dot\gamma[~t~]$ is called the tangent vector of curve $\gamma$, which is characterized by

*its direction (i.e. the equivalence class of all curves which are just touching the image of curve $\gamma$ at event $\gamma[~t~]$), and

*with magnitude such that the definitive equation shown below is satisfied.

But observers should also be able to measure length, right?

Well, surely it should be possible to attribute arc length also to curves which are (throughout) space-like. At least formally, that's straightforward, too (using the same sign convention as above):
$$L[~\gamma~] := \int_0^1~dt~\sqrt{-g[~\dot\gamma, \dot\gamma~]},$$
where here the curve $\gamma$ is understood and required to be space-like;
and, as above, the magnitudes of applicable tangent vectors are given only through a corresponding definitive equation shown below.

the first equations is the one defining [...]

The two equations shown above certainly don't provide self-standing definitions since they depend on suitable assignments for the magnitudes of tangent vectors; perhaps related to a particular ("good", "affine", "locally orthonormal") assignments of coordinates.
Definitive are rather expressions of arc lengths (of space-like curves or of time-like curves, respectively) in terms of given


*

*distances $d : \mathcal S \times \mathcal S \rightarrow \mathbb R$, or

*Lorentzian distances $\ell : \mathcal S \times \mathcal S \rightarrow \mathbb R$,
as
$$ \Large{ L[~\gamma~] := \text{lim}_{\{\frac{d[~\gamma[~p_{(k)}~], \gamma[~p_{(k + 1)}~]~]}{\text{Max}[~\{~d[~\gamma[~p_{(a)}~], \gamma[~p_{(b)}~]~]~\}~]} ~\rightarrow~ 0\}} [~\sum_{k = 0}^{n - 1} d[~\gamma[~p_{(k)}~], \gamma[~p_{(k + 1)}~]~]~]} $$
and
$$ \Large{\tau A_J^Q := \text{lim}_{\{\frac{\ell[~\gamma[~p_{(k)}~], \gamma[~p_{(k + 1)}~]~]}{\text{Max}[~\{~\ell[~\gamma[~p_{(a)}~], \gamma[~p_{(b)}~]~]~\}~]} ~\rightarrow~ 0\}}~[~\sum_{k = 0}^{n - 1} \ell[~\gamma[~p_{(k)}~], \gamma[~p_{(k + 1)}~]~]~]} $$
respectively, where $p := \{p_{(k)}\}$ is any suitable partition of interval $[0 ... 1]$.
A: Exactly as an ideal clock at rest with the observer (here pictured as a timelike curve) measures the proper time of the observer, ideal rulers  at rest with the observer measure the distances in the rest space of the observer. Mathematically these rulers are pictured as an orthonormal basis made of $3$ vectors normal to the unit tangent vector to the timelike curve describing the observer. 
As they are orthogonal to that timelike vector, they are spacelike and the Lorentzian scalar product restricted to the $3$-space spanned by them is positive, i.e., Euclidean. Moreover, orthogonality implies that the speed of light rays is $1$ (I am working with the convention $c=1$) if the observer uses these rulers and the ideal clock at rest with him/her to measure that speed.   
An interesting question concerns how these rulers are transported along the curve. A natural choice is the parallel transport. However, Fermi-Walker transport is also another possible way (both preserve orthogonality an metrical  properties and they coincide if the curve is a geodesic).
Everything I wrote concerns the "infinitesimal" rest space with the observer (pictured in the tangent space of an event crossed by de curve). If you whish to have a finite or global notion, you should collect several observers: a smooth congruence of timelike curves. A $3$-manifold tangent to all  "infinitesimal" rest spaces of the observers is a global $3$-space at given time.  A smooth Euclidean $3$-metric turns out to be defined automatically thereon from the positive scalar products in each infinitesimal rest space of the observers.
Sometimes no such $3$-manifolds exist for a given congruence of observers,  as it happens, in particular for the rotating global platform. In that case, a more sophisticated notion of rest space is necessary (and possible). 
A: In GR the notions of space and time are no longer different, they are combined into spacetime. Points in spacetime are labelled by coordinates, which can be arbitrarily chosen, and can in the special case of Minkowski space be the familiar x, y, z and t. The only physically meaningful quantity however is the line element:
$$ds^2 = g_{\mu \nu} dx^\mu dx^\nu$$
where $g_{\mu\nu}$ is the metric. The proper time along a time-like curve is the integral of the line element, as you've read. I should stress however that this is not the same as coordinate time, although they converge in the Newtonian limit.
Measuring distances in GR is more subtle, and most importantly there is no unique notion of distance. You could mean the line element at t=constant, and integrate it along some space-like curve. You could make up some sort of measure of distance involving light, cosmologists for example commonly use luminosity distance and angular distance, which has to do with how bright and large luminous objects appear in the sky.
In conclusion there is no unique equation for "distance".
