Just checked one job, all it says is, that the applicant must be able to obtain ITAR accreditation from the Department of State. Nothing really extraordinary in aerospace, especially for an US company. Weird so, that SoaceX is emphasizong the ITAR aspect so much, but then I have no experience how that is handled in the US, maybe that normal.
That being said, ITAR is not a job classification, ITAR covers the move of anything defense or dual use related (data, information, hardware...) outside the US with some weird things about nationalities of people handling those things. It has nothing to do with classified or secret information, that is something different in addition to ITAR.
The thing that puzzles me about roles like janitors: ITAR is "need to know, right to know", meaning if you don't have both, you don't need access to ITAR protected data. And since the normal janitor, or payroll admin, has neither nobody bothers those roles with ITAR normally...