You mean do strobe, strobe, strobe, strobe, pause, pause, pause, pause? I bet that's at least as bad as holding the source on for the first four intervals and then off for the latter four intervals.

In any case, if you actually have a scene bright for 1/24th of a second and then dark for 1/24th of a second, repeating, you're well within photosensitive epilepsy range. Don't do that to your actors unless you've discussed it with them and with your insurance company first.

So, shoot at 240 fps and strobe set lights for 1/240s and backdrop for 1/240s.