A mayor requires an Office (a room created from a chair/throne, whatever else is in there) and a dining room (defined from a table, whatever else is in there) and a quarters (bedroom, defined from a bed, ditto). Is that last room correct? Or did you try to create a 'quarters' by the additional furniture (chests, weapon-racks, armor-stands) alone? There is no requirement for a personal barracks (you can't even assign a barracks as a 'personal room', though you can make it serve a 'unit of one'... For no relevence to noble demands).
And, after all that, the rooms must be of a given quality level (you should, on the Nobles screen see whether the quality needed/given mismatches, up or down). And if you overlap rooms (putting the base-furniture in one area and letting their extent of influence overlap each other) the quality of a given room is reduced by the nature of the overlap. This might also be why a Barracks is causing problems with a Bedroom.
I tend to cut out four rooms as a 'suite'. Typically.a square arrangement of 5x5 rooms, with a doorway into one of them (sometimes the office, sometimes the dining room, depending on if the Mayor is also in another office-using noble-role that would appreciate easier office access, rather than quicker dining experience) and 'internal' doors (within the block, that is, not official "internal doors", which is another thing entitely) to the adjacent rooms that are the counterpart dining/office room and the bedroom. The fourth room (a further door through from one the others, opposite corner to the entryway) I set up as potential/actual Tomb room from a sarcophagus, despite a Mayor not yet needing one. At some point in the future someone will ask for one.
The chests, etc, I then dot around the suite to try to up the quality of the rooms that need better total quality levels. And there must be (non-internal!) doors, often, to ensure the room designation is maximal wihout flood-filling over into an adjacent area.
To up the quality further, where still lacking, I then start to smooth the floors and walls (or continue, as I'll often start the smoothing on and around the chair/table/bed/door/etc placement points, as a vusual reminder) and that tends to be more than sufficient to satisfy the noble requirements at this level. Once you get upgraded titles you're going to want to ensure you have masterwork furniture (maybe of a 'favoured' or just intrinsically valuable source material) ready to populate those future (or regfurnished) suits.