CuriousOne has, as he/she/it is wont to do, provided a perfectly satisfactory answer as a comment. However just for completeness let's provide the details as an answer.
In general relativity the length of any curve is calculated by integrating the line element $ds$ along it, where $ds$ is given by:
$$ ds^2 = g_{ab}x^ax^b \tag{1} $$
In this equation $g$ is the metric tensor, so without a metric tensor there is no way to calculate the length of a curve. Since the length of an object is just the length of the (straight) curve joining its ends, without a metric tensor we cannot measure the length of objects.
But even in the absence of a metric, we can determine if two objects are the same length by superposing them and seeing if their ends coincide. To do this we have to parallel transport one or both of the objects to sperpose them.
However it should be pointed out that there is no guarantee that parallel transporting an object will preserve its length. This is the case in general relativity, but for example it is not the case in Weyl's proposed theory for unifying gravity and electromagnetism. In the article I link see 7. Problems with Weyl's Theory starting at page seven.
Response to comment:
The raised indices in equation (1) aren't powers. They just identify the components of a four-vector and run from $0$ to $3$.
$$ \vec{x} = \left( x^0, x^1, x^2, x^3 \right) $$
This is the sort of thing you take for granted when you're used to this stuff, and it's easy to forget newcomers to general relativity get confused. Sorry!
The notation I've used was invented by Einstein to make differential geometry a bit less verbose. The $a$ and $b$ are just indices and when an index is repeated it is summed over. So the equation I've written is shorthand for:
$$ ds^2 = \sum\limits_{a = 0}^{3} \sum\limits_{b = 0}^{3} g_{ab}x^ax^b $$
