Can an E5 have an E4 roommate in the barracks, who are in the same company (separate rooms, shared bathroom)?

Don't know about now, but back in my day, there was no way this was even considered. NCO's had their own floor/private rooms away from the enlisted. Only person allowed into that area was CQ/Staff Duty & Officers. You didn't want to be caught dead in that area if you wanted to not do PT for the next millenia.

Huh What? Hmmm, Lets see as an E-6 on my first ship I shared a Berthing Compartment with 30 other guys E-3 thru E-6 and on my Second it was 15 other guys E-4 thru E-6. In a Navy Barracks Room I don't remember there being any Separation between an E-4 Petty Officer 3rd Class and E-5 Petty Officer 2nd Class now once you made First Class of course there was only two to a room and it was always both E-6s Petty Officer First Classes. I did get a kick out of when I checked into RAF Lakenheath for a Month. "Petty Officer Nagel we are real sorry but you will have to share a Kitchen with another room/individual" "Lady I'm used to sharing a whole lot more".

Let me ask the question this way.
If the NCO and Specialist were both Geo-Bachelors would the unit be okay with you getting an apartment together? Hopefully I would think they would be against that, and if that is not okay I don't think a Specialist and an NCO should be sharing barracks. Only exception I would agree with is if resources are limited to that extreme and that Specialist was in a Team Leader position.