Quirky Household

That was this family's specialty: strange conversations.

—Tomoya to Mei Sunohara, regarding the Furukawa household following a conversation about who Nagisa would allow to check for a nonexistent scar on her butt, Clannad

Now remember, as far as anyone knows, we're a perfectly normal family...

An ensemble of bizarre characters who are related, or might as well be. Unlike the Dysfunctional Family, we as the audience plainly see the family is extremely well adjusted, supportive and loving—more so than some "real" families. They are also easily able to absorb friends, acquaintences, and distant relatives into their structure.

This is coupled with a range of quirks easily labeled "bizarre" by any of their peers.

It's also very convenient for heroes to have these, as they're not bound by the Masquerade, weirdness is normal.

Arguably started with the Oscar-winning play/film, You Can't Take it With You, and The Addams Family on television. Later supplanted by the trend of "realistic" but dysfunctional families (except in Anime), probably as a Deconstruction of the traditional family system.

Arguably part of the natural growth of a Pretty Freeloaders group. For a larger grouping, see Quirky Town.

Examples of Quirky Household include:

Anime and Manga

The Kurata family in Kodomo no Omocha.
The Paper Sisters and the famous writer Nenene from the tv series of Read or Die basically adopted each other.
The Kawai family from the Pretty Sammy series (Alternate Universe #276 of the Tenchi Muyo! multiverse). Her parents were so insane but easygoing, she didn't need Parental Abandonment to have the freedom to run around fighting monsters.
The residents of Aoi House.
Yuu and Miki's parents in Marmalade Boy met, liked each other so much that they divorced and remarried each others' spouses, and then all moved in together into one big house.
The Shinju-yu (manga)/Pearl Piari (anime) from Mermaid Melody Pichi Pichi Pitch, where the main trio lives with their Weasel Mascot and Cool Big Sis.
Meimi and her parents in Kaitou Saint Tail.
Both the household of the Takamachis and the Harlaowns of Magical Girl Lyrical Nanoha.
- For that matter, the Yagami household. It consists of a crippled girl and her humanoid program Guardian Knights, the Wolkenritter.
- The Nakajima clan ain't no slouch in this regard, either.
Yuan's family in Samurai Deeper Kyo. Also, the Shiseiten, if you look at them as a family and not True Companions.
The Hiiragizawa household in Cardcaptor Sakura. The "dad" looks like a ten-year-old kid, and his "children" are a genderless magical being posing as a teenage schoolgirl and a winged cat that looks like a toy most of the time.
The main cast of Kyouran Kazoku Nikki is the perfect example of this.
Although with only two members, they don't make as dramatic a showing as some examples listed above, arguably the Koiwais from Yotsuba&!. Certainly, in his way Dad is nearly as quirky as Yotsuba, if more laid-back about it.
The Furukawa family from Clannad featuring a kind mother who is a lethal baker, a Hot Dad who is still a kid inside and a shy but cute daughter who's somewhat of an airhead.
In Katekyo Hitman Reborn, one by one Reborn, Lambo, Bianchi, I-Pin, and Futa all freeload off of Tsuna.

Comic Books

The Crosovers, who buck the idea that the Quirky Household residents don't need to worry about The Masquerade, because every member of the family is maintaining a different Masquerade. Dad's a Flying Brick superhero, mom's a vampire hunter, the Bratty Teenage Daughter is a Sword and Sorcery heroine, and her kid brother is communicating with The Greys.

Film

The titular Robinsons from the Disney movie Meet the Robinsons.
In Unstrung Heroes: young Steven Lidz' home, to a lesser extent. Especially quirky, however: the labyrinthine apartment (filled with hoarded junk) in which he lives with his eccentric uncles for awhile, after his mother's illness exacerbates the awkwardness of Steven's relationship with his father. Franz—actually Steven, now rechristened by his uncles; It Makes Sense in Context—undergoes an disorienting yet often enjoyable identity crisis. Then, as it turns out, at least one of his uncles is a bit more than merely eccentric. Franz must navigate between his uncles' pride in their heritage (complicated by imagined anti-Semitism lurking everywhere), and his inventor father's 100% materialist (i.e. anti-spiritual) worldview.

Literature

The Moomin family, from the childrens' book series by Tove Jansson.
Roleplay example—the cafe in Kokoro.
The Bagthorpe family in the Bagthorpe Saga.
The Cassons of Saffy's Angel and it's sequels.
The Weasleys arguably fit this, due to their seven children (nearly all with extremely different personalities), Mama Bear mother, eccentric father, weird pets (hyperactive owl, ancient, infeebled owl, and ghoul), all crammed into a small, ramshackle house and, oh yeah, they're all wizards.
- If they qualify, then they're normal compared to the Lovegood Household.

Role Playing Games

There Is No Such Thing as Notability—Often in multifandom Role Playing Games on LiveJournal, characters will form together in a (very quirky) band of True Companions depending on where they live. For example, the game Polychromatic has characters from Princess Tutu, Ouran High School Host Club, Chrono Crusade, and Count Cain among others that have settled in a building known as "The Opera House". The result is a chaotic but tightly-knit group of character that often treat each other like family. (Poly has Loads and Loads of Characters, so this is just one example of many.)

Television

Subverted by The Young Ones, in which none of the characters in the house can actually stand one another for any significant length of time. This doesn't count as an instance of Dysfunctional Family, as the characters in a Dysfunctional Family show are permitted to get along with one another despite their differences.
The Coneheads from Saturday Night Live.
The Addams Family was mentioned above, but their knockoffs The Munsters (and The Gruesomes on The Flintstones) qualify as well.
Tom, Dick, Harry and Sally. On Earth.

Theater

Made famous in the 1936 Pulitzer Prize-winning Broadway theatre hit, You Can't Take It with You, by George S. Kaufman and Moss Hart, making this a fairly well aged ensemble. This also applies to the movie verson.

Web Comics

At many points, the cast of Sluggy Freelance.
Bob, Jean, Molly, and Snookums in The Inexplicable Adventures of Bob. Could probably expand it to include Galatea, Auntie Voluptua, and Djali.
The Verres family of El Goonish Shive, consisting of two shapeshifters (One an adopted weresquirrel with enormous magical power, the other a ChivalrousPerverted Mad Scientist who causes oodles of Gender Bending.) is the most noticeable example. Far more subtly is the Dunkel household which is fairly normal other than having an Opposite Gender Clone of the Ordinary High School Student Elliott.

Dan: I also liked that reasoning because it gave me the idea for this comic, and I love writing scenes like this. There's a certain madness to the Dunkel household that, in my opinion, makes the Verres household look relatively sane. It's enough that I feel I must now assure you, the audience, that there are no questionable ingredients in Mrs. Dunkel's brownies.

- To clarify, the Dunkels show almost no concern (compared to most people) when their son randomly changes sex or breaks into a government installation and bringing back a mysterious twin sister. They just give a lecture then offer their kids brownies. It gets really, really weird after a while.
The River family from Irregular Elis. A spanish webcomic about a Badass Family of "Superheroes" with a lot of Hanna-Barbera influence.

Web Original

Agents of the Protectors of the Plot Continuum tend to come in pairs, and while antagonism between them is sometimes played up for comedic effect, this trope applies almost universally

Western Animation

The Planet Express staff from Futurama straddle the border between the Quirky Household and True Companions. In the episode "Future Stock", Fry even says, "We're not a traditional family, like the Johnsons next door or the lesbian coven across the street, but we're still a family!"
The boarders in Hey Arnold! are portrayed this way in several of the later episodes.
The residents of Foster's Home for Imaginary Friends.
The Oblongs. 'Nuff said.
The Chan family, in Jackie Chan Adventures, have a rather strong sense of this, consisting of a twelve year old who can sneak in anywhere with almost ninja-like efficiency (whether she's wanted or not), a grumpy, snarky old witchdoctor, a former sumo wrestler turned villianous Tank, turned Gentle Giant chi wizard in training and a rather stressed out archeologist with a knowledge of martial arts as good as... well, he's Jackie Chan, you do the math.
The Flyn-Fletcher Family from Phineas and Ferb. A father whose an expert on random and obscure antiques, a mother who was once a one-hit-wonder, a semi-neurotic teenage girl who likes screaming at cheese, a pet platypus whose secretly a special agent, and two brothers who do everything.