It goes like this: federal law prohibits firearms on all post office property, including parking lots. The prohibition applies whether or not any postal employees are present, whether or not the post office building is attached to the parking lot itself (or is on an adjacent lot), or whether any actual security is provided.
In addition, as Mr. Kopel points out, the U.S.P.S. is a monopoly by congressional statute, and while other services (e.g. U.P.S. and FedEx) and methods of communication (e.g. e-mail, text, long-distance phone calls) have become viable alternatives to traditional mail service, nearly every person in America will have to use the postal service for something, simply because there are no other feasible options, or because it's not their choice; they're just picking up a letter/package someone else sent.
In Bonidy, the challenge stems from the situation in the plaintiff's town, Avon, Colorado. There's no home mail delivery in Avon; it's too spread and rural for home delivery to be practical. Residents, therefore, are required to access their mail by going into the post office building, which is only staffed six hours per day — the lobby is open 24 hours per day and has nothing in place resembling "security". As it stands, the current ruling from District Court Judge Richard Matsch is that the post office lobby is among the "sensitive places" provided for in Heller (where gun bans are allowable), but the parking lot is not. Mr. Bonidy can leave his gun in his car while picking up his mail.
So what makes this applicable to me, personally?
I work at **mumble-government-agency-mumble**, where firearms (along with a rather extensive list of "weapons" in general) are banned, per a state law that also bans them in state-owned parking lots — employees with CCW licenses can't even leave the gun in the car. My building is legally "secured"*, and so would probably pass the "sensitive place" test.
The parking lots, however, are another story.
The two state-owned lots nearby (serving several agencies in several buildings) both occupy a full city block (i.e. streets on all four sides) and neither has any permanent structure on it. The lots cannot by any reasonable definition be considered "sensitive places" and therefore subject to prohibitions on carry, but because they are state-owned property, the carry ban stands. The nearest public parking garage is nearly a mile away, and while privately-owned parking areas are available, they're permit-based (expensive, always full, get on the waiting list, etc.), the rules vary, and they're not much closer than the public parking. In short, the state-owned lots are the most feasible for the most people.
If Bonidy is upheld, there's hope that CCW-licensed employees — even if they must be disarmed while at work; a challenge for another time — could legally carry while in transit and store their guns in their vehicles if they use these lots. The parking garages below some of the buildings themselves, being literally under a "sensitive place", could reasonably fall under the "sensitive place" exemption. But progress is progress.
(Hat tip goes to Knoxville Gun Rights Examiner Liston Matthews, whose article I found via David Codrea's blog, "The War on Guns".)
* - "Secured", as in there's a receptionist checking badges/IDs at the main entrance; other entrances are accessible by swiping a badge. They're all glass doors, no barrier to someone willing to make a little noise to gain illicit entry. There's no consistent security personnel; one unarmed guard is posted at each of three entrances if an employee reports a specific threat, but there are a lot more than three entrances. Personally, I can't with any honesty consider this "secure", but the law and I disagree.