Great question!
First re gunshop: NO. Gunshops in VA and MD are required to have an FFL (license). That license requires the owner to "log in" any firearm on the presense for more than 24 hours. The rule is: more than 24 hours = "transfer." Shops do not offer a "storage option" and even if they did, they would have to price it as high as their transfer fee (about $30 around here). But, as is, they just do not want to mess with it. What if you do not return for the gun? Just too much of a risk for the shops. A shop where I once helped out used to offer a "gun cleaning service" through the gunsmith we had - but I just called Sharpshooters for you & they do not offer such a service. I doubt others would.
Second: You CANNOT visit D.C. with a gun. Or ammunition. Or even a fired shell casing. Having any of these with you - even accidentally - is a crime in D.C. ("unregistered ammunition").
In theory, a federal law called FOPA might protect you but ONLY IF you just drive straight through the ONE interstate in D.C. (route 295) with a gun unloaded in the trunk, but only if you did not stop and were only passing through. My shooting buddies who work as police in D.C. have consistently told me that if they catch you on ANY road other than the interstate in D.C. with any gun, you will go to jail - even if you are just passing through (ie FOPA does not apply anywhere in D.C. but 295. That interpretation of FOPA may be wrong, but that is the view of the police in D.C. and the courts have backed them up on it (as far as I can tell).
However, you specified "visit." No way to do that with the gun in D.C.
So what do people do? I have read here and on other gunforums that people sometimes leave the guns at friends homes in VA or MD (no, I will not do that for anyone). If the gun is locked in a case that the other person does not have access to, it might be OK; I am sure the issue is debatable.
I also know hunters will sometimes mail a rifle or shotgun to themselves at a hotel where they plan to stay - and the hotel staff is not in violation of law for receiving an unopened package for a guest.