One can construct a square with sides of a given length $s$ using compass and straightedge as follows:

1. Draw a line segment of length s. Label its endpoints $P$ and $Q$ .
   
   
   
2. Extend the line segment past $Q$ .
   
   
   
3. Erect the perpendicular to $\overrightarrow{PQ}$ at $Q$ .
   
   
   
4. Using the line drawn in the previous step, mark off a line segment of length $s$ such that one of its endpoints is $Q$ . Label the other endpoint as $R$ .
   
   
   
5. Draw an arc of the circle with center $P$ and radius $\overline{PQ}$ .
   
   
   
6. Draw an arc of the circle with center $R$ and radius $\overline{QR}$ to find the point $S$ where it intersects the arc from the previous step such that $S \neq Q$ .
   
   
   
7. Draw the square $PQRS$ .
   
   
   

This construction is justified because $PS=PQ=QR=QS$ , yielding that $PQRS$ is a rhombus. Since $\angle PQR$ is a right angle, it follows that $PQRS$ is a square.

If you are interested in seeing the rules for compass and straightedge constructions, click on the link provided.