18 USC 930 says:
"(a) Except as provided in subsection (d), whoever knowingly possesses or causes to be present a firearm or other dangerous weapon in a Federal facility (other than a Federal court facility), or attempts to do so, shall be fined under this title or imprisoned not more than 1 year, or both. "
(d) Subsection (a) shall not apply to—
(3) the lawful carrying of firearms or other dangerous weapons in a Federal facility incident to hunting or other lawful purposes.
"(g) As used in this section:
(1) The term “Federal facility” means a building or part thereof owned or leased by the Federal Government, where Federal employees are regularly present for the purpose of performing their official duties. "
"(h) Notice of the provisions of subsections (a) and (b) shall be posted conspicuously at each public entrance to each Federal facility, and notice of subsection (e) shall be posted conspicuously at each public entrance to each Federal court facility, and no person shall be convicted of an offense under subsection (a) or (e) with respect to a Federal facility if such notice is not so posted at such facility, unless such person had actual notice of subsection (a) or (e), as the case may be."
So, the building or a part of the building has to be leased by the Feds, the employees have to be Federal, and notice must be posted. If all 3 of those are true, you then get to argue whether or not your carrying falls under "other lawful purposes". That would probably fail because the Post Office also has federal regulations, one of which is 39 CFR 232.1, which states:
"(l) Weapons and explosives. No person while on postal property may carry firearms, other dangerous or deadly weapons, or explosives, either openly or concealed, or store the same on postal property, except for official purposes"
Again, here you get to argue "official purposes" and I would guess that means mailing a firearm. The USPS generally prohibits mailing handguns unless from a dealer or manufacturer. Rifles and shotguns can be mailed. More information on that can be found here: http://pe.usps.gov/text/pub52/pub52c...PSk741320edwa.
So you'd most likely lose that argument with a handgun.
Having said all of that, does this "Post Office" post that notice and are the employees being paid by the Feds? If not, it should not be considered a federal facility. Then you have to figure out whether or not 39 CFR 232.1 applies. The Code of Federal Regulations is designed to implement the laws set forth in the USC in minute detail. The CFR are treated as law, provided they are a "reasonable interpretation" of the statutes. If it is not a federal facility, you would think that 39 CFR 232.1 doesn't apply.
But just for grins, let's say that you did carry there and were arrested. You can't be convicted unless that notice was posted. That would be my litmus test: No sign, no prohibition.