Most boats that will fit six people are going to be significantly longer than a car, especially if they're on a trailer with a long y-tongue sticking out front.
Even if you're planning on only daysailing, you'll probably want a boat with a small cabin... since you could get caught out in bad weather, and not having a cabin will make the wife and kids very unhappy in a sudden rainstorm/squall.
It would also help if you said what your budget is...
One boat I can think of that would be pretty much perfect for you is the Farrier Tramp/Eagle. It's only 19' long, easily trailered, relatively light, very shoal draft... but finding one for sail might be tough. A Corsair 24 would also be a really good choice.

But I'm a bit biased in favor of trimarans.

The shoal draft and trailerability can be problems with monohulls.
Compac makes a series of boats that would also fit the bill. The Compac Suncat, (a cat boat not a catamaran), or the larger of the traditional catboats from Marshall.
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Sailingdog
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Telstar 28
New England
You know what the first rule of sailing is? ...Love. You can learn all the math in the 'verse, but you take
a boat to the sea you don't love, she'll shake you off just as sure as the turning of the worlds. Love keeps
her going when she oughta fall down, tells you she's hurting 'fore she keens. Makes her a home.
—Cpt. Mal Reynolds, Serenity (edited)
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