As for real world applications, most of the buildings around you older than 30 years were designed by an architect that manually used geometric constructions for their drawings. I learned geometric constructions in my Tech Drawing class before I learned them in maths class. Of course, CAD programs do all of that in the background now, but the reason that Euclid chose the axioms that he did for his constructions was that it was about the real world. Otherwise, why include the parallel postulate?
So as for a story, how about using a compass and straightedge (no rulers if you want to be pure) to design a net of an interesting building (ie. not a cube). Starting from a single length to define the height of a wall, you could build a gabled square house, a hexagonal round(ish) house, or a geodesic dome (soccer ball) all from simple geometric constructions. You would end up with a little Euclidean village.
If you have studied symmetry, the structures could be decorated with various wallpaper and frieze patterns, maybe replicated with similar techniques. One of your students could invent a recursive/fractal wallpaper!
If you have the opportunity, using software like Geogebra can extend the scope of design in purely geometric terms. The infinite scalability from changing the first and only measurement is also amazing to watch. A revolutionary tool for any part of geometry if you have computers or tablets available.
Though not using Euclidean constructions, I had some able students create nets for every Platonic and Archimedean solid (and samples from the two infinite series of prisms and skew prisms) in Geogebra, discovering Euler's theorem for plane networks and then using that to discover the more difficult combinations. A couple of the most complicated Archimedean solids were hard to lay out and they needed a helping hand, but it was a worthwhile activity, though it takes some time.
Constructions can be very rich, especially when combined with other areas of mathematics. I hope your students have fun with whatever you choose.