What can I do if my paper doesn't fit any arXiv category?
The archive engrXiv is dedicated to engineering. Wikipedia has a list of preprint servers where you might find other possibilities, but the engrXiv is the only one dedicated to engineering, as far as I can see.

There are other solutions to publish preprints as well: your institutional repository or generic archives such as figshare or Zenodo. Both accept articles and will provide a DOI.