For compass and straightedge problems, you could have a line between two points A and B, and want to make a line the same size between C and line DE.
If you placed the two points of the compass between A and B, and made a circle around C with the same radius, that would achieve this result.
But is this something you are allowed to do?
On
Putting aside the specific construction of this query: The general rule in strictly classical Euclidean constructions is: The instruments cannot carry distance information from one step to another. So, in particular: (a) The straightedge cannot be marked, not even at a single point; (b) The compass does retain its spacing while drawing any one circle, but forgets the spacing once the compass is 'lifted off of the plane'. If you want to replicate a distance at a specific locale, you generally need auxiliary constructions (e.g parallel lines through a segment's endpoints).
Yes. Not by the rules about how to use compass and straightedge but because it can be proved that it's as if we could do it (that's proposition 2 of book I of Euclid's Elements).