It all depends on your situation...
If you're out in the boondocks, with no neighbors nearby and are known to own a lot of high-dollar gear that professionals might be interested in stealing, then you're going to need to go to a lot more trouble than if you're in town with enough surveillence by neighbors that thieves would only have a few minutes to B&E.
The "textbook" answer for an arms room says GSA-approved safe door and lock, reinforced concrete or brick walls, reinforced concrete floor and ceiling. That's overkill for a lot of folks, and maybe not enough for some. See Army FM 19-3 (19-30?) "Physical Security" for some good info.
In a typical house or apartment there are three weak points: the door and frame, the windows (if any), and the walls.
Installing a heavy duty metal door, or a safe door, does no good if the door frame is weak enough that the entire door assembly can be torn out quickly.
Windows are a problem and need to be blocked in some fashion. Removing them completely is an option, but may can make it obvious that you have a secure room. Closing the window off on the interior and leaving a "display" window visibile from the outside is one way to handle this. Interior bars are another option.
The walls are the major problem. At a minimum, they need to be reinforced to prevent a burglar from merely kicking through sheetrock to access the safe room. One option is to use a couple of layers of plywood, with sheetmetal between the layers, to form an inner liner for the room. Tied in, of course, to the hardened door frame. That will make it harder for a burglar to simply saw a new door in the wall with a circular saw.
The key is to use layers of security, starting at the kerb: good landscaping and lighting to encourage a burglar to pick another house, good relations with the neighbors so they'll notice and do something if someone is tampering with your house, good doors, windows and locks on the exterior to make it hard for thieves to enter the house, a good door, frame and locking system to make it hard to enter the arms room, good walls to make it hard to bypass the door, an alarm system to alert everyone that someone who doesn't belong is in the house, locking racks inside the arms room to make it hard for a thief to grab and go once he's inside...
Beyond the security issues, there's the environmental issue. Make sure it's well ventilated. You may need a dehumidifier in some places. It's not likely to flood after a bad rainstorm, no roof leaks, etc. If you're planning on storing ammunition in the room, is it cool enough in the summer (or air conditioned) so that your stockpile won't be baking at 90F or higher for months?
It's all much easier to do if you have the chance to build a proper arms room while the house is being built. Most of us, of course, will have to make do and retrofit.