If you use this overpass, you’ll get the streets in the town (Leuven in my example) you list in the code.
[out:csv("name";false)];
area[name="Leuven"];
way(area)[highway][name];
out;
For Leuven I got almost 28000 names back.
I wouldn’t try it for the whole country, possibly leading to memory problems.
Please note that you have to filter out all the duplicates yourself…
If a street has more than one (named) segment, you’ll get that name for every segment in the result.