The first Death Star had a diameter of between 140 and 160 kilometers. The second Death Star’s diameter ranged from 160 to 900 kilometers.
There are two near term technologies which could be applied to making Death Star sized structures:
1. Space bubbles
2. Robotic spiderfab construction
A single bubble can be 10 meters in earth gravity, 100 kilometer in low earth orbit or 1000 kilometers in deep space. Foams made of many bubbles could be far larger in size.
A 140 kilometer diameter sphere would have 61600 square kilometers of surface area. If a structure could be made that was one gram per square meter, then that bubble sphere would be 61600 tons.
Reusable Spacex falcon heavy launchers could bring up the material to expand a large bubble in space. It would take about 2000 launches of a Spacex Falcon heavy to place the amount of material in orbit for a 140 kilometer sphere if the mass budget of one gram per square meter could be maintained. Every additional gram per square meter would be another 2000 launches
The Nazi considered making giant mirrors to heat cities to several hundred degrees. A Death Star size hollow space sphere would be to concentrate light 100 times over an area 14 kilometers across. So while it would not destroy a planet, it could provide electricity for 100 cities or cause one city to burn.