Except for an FFL shipping pursuant to regulation, it is a heap big no-no (technical legal term for a felony) to have a handgun anywhere and in any fashion on Post Office property, even in the parking lot.
It is
not a felony to merely possess a firearm on Post Office property. It is an infraction that could get you 30 days in jail and a $5,000 fine. 39 CFR 232.1:
Title 39: Postal Service
PART 232—CONDUCT ON POSTAL PROPERTY
§ 232.1 Conduct on postal property.
(a) Applicability. This section applies to all real property under the charge and control of the Postal Service, to all tenant agencies, and to all persons entering in or on such property. This section shall be posted and kept posted at a conspicuous place on all such property. This section shall not apply to—
(i) Any portions of real property, owned or leased by the Postal Service, that are leased or subleased by the Postal Service to private tenants for their exclusive use;
(l) Weapons and explosives . Notwithstanding the provisions of any other law, rule or regulation, 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.
(p) Penalties and other law. (1) Alleged violations of these rules and regulations are heard, and the penalties prescribed herein are imposed, either in a Federal district court or by a Federal magistrate in accordance with applicable court rules. Questions regarding such rules should be directed to the regional counsel for the region involved.
(2) Whoever shall be found guilty of violating the rules and regulations in this section while on property under the charge and control of the Postal Service is subject to a fine as provided in 18 U.S.C. 3571 or imprisonment of not more than 30 days, or both. Nothing contained in these rules and regulations shall be construed to abrogate any other Federal laws or regulations or any State and local laws and regulations applicable to any area in which the property is situated.
18 USC 3571:
18 USC § 3571 - Sentence of fine
(a) In General.— A defendant who has been found guilty of an offense may be sentenced to pay a fine.
(b) Fines for Individuals.— Except as provided in subsection (e) of this section, an individual who has been found guilty of an offense may be fined not more than the greatest of—
(1) the amount specified in the law setting forth the offense;
(2) the applicable amount under subsection (d) of this section;
(3) for a felony, not more than $250,000;
(4) for a misdemeanor resulting in death, not more than $250,000;
(5) for a Class A misdemeanor that does not result in death, not more than $100,000;
(6) for a Class B or C misdemeanor that does not result in death, not more than $5,000; or
(7) for an infraction, not more than $5,000.