Walter, that's my old neighborhood, so you have the right person.
If you look at the map, you can see that to the north and south, there are parks. to the east and west are buildings. The land falls away at a pretty significant grade that runs roughly parallel to Pennsylvania down to constitution and the buildings follow this, resulting in some weird architecture. Constitution used to be a canal (never terribly successful) that went from I think where Rock Creek hit the Potomac to in front of capitol hill, so consider that might have filled in post apoc. Things that would be interesting in a feral DC would be metro stations (well established undergrounds meeting street access) and some of the landscape. The federal bits downtown are built on a river valley that goes up as you go north generally. the post apoc would have DC return possibly to a marshy area, but not as high up as the White house.
In the immediate area, of the white house, you have the treasury building to the east and the old executive office building to the west. Beyond that, east and west and north it's all office buildings that are pretty generic but - and this requires emphasis - NO BUILDING IS HIGHER THAN THE CAPITOL DOME ANYWHERE IN DC. If you have a skyscraper, that's totally wrong. I think the highest you can go is 14 stories. Also, there are places where there are laws that keep buildings low to preserve historic sight lines. I used to live in Logan Circle, so I'm aware particularly of that one, but I am sure there are others.
All this said, the White House would be an absolutely terrible squat from a security standpoint. It's not that big. Every other building in the vicinity looks down on it with the exception of some of the historic places like the Blair house across the street. Sure, there might be a very cool bunker in it, but I'd think that the Phone Booth (The Verizon Center, where the Caps play Hockey) would be a more interesting gang turf. Access to the underground, big area attached to another big area, would be more interesting. Plus, it's got the Chinatown arch.
If you really wanted to have a cool place that's near the White House, try the National Press Building. It's hollow on the inside and has a mall that runs the whole block and has entrances on three different levels on three different sides of the block (working from memory, but I think that's right). It's also attached to a large office building for space (I used to work there). It's on that rise I told you about earlier.
Couple of things to think about when considering a post-civilization DC:
There is a humongous park that cuts upper northwest DC in two and divides places to the north of the white house from georgetown and places north from there. There are a few places where that gets narrow - it's generally just a crossing until you get north of Mass, but there is a significant creek that runs through it. I'm currently a few blocks from one of the boundaries on 16th street and I've had deer poop in my yard.
DC has significant terrain, being in a river valley. Ignoring that would stand out. It's not like this is san francisco, but if you have some guy just running up 16th from U like it's no big deal and he's not a superhero, you made a mistake.
DC has row houses, but they aren't as big as the brownstones in harlem and they certainly aren't tenements. DC is majority black but the distribution is not uniform. As a rule, the further west you go, the more white the people are, but that is a VERY loose rule. The Adams Morgan area is, or has been hispanic. The biggest rule of DC is that it's very culturally mixed. And if you can mix in a mention of Chuck Brown and the Go-Go beat in, you will impress the natives. I firmly believe that the Go-Go beat will survive the apocalypse as it is most often performed on pickle barrels and flipped over trashcans anyway.
Hope this helps or is in some way amusing.