Looking for a Unicorn

Looking for a unicorn house rental:

4+ beds
2000+sqyds
Fenced yard
High School rating 6+
Not dated from the 70s
$5000/mo max

We have 3 kids, 2 large dogs (60+ lbs), 1 cat

I work at Pentagon and Navy Yard. Commute<1hr.

Looking in Burke, Lorton, West Springfield, Vienna, Fairfax,McLean/Langley, Rockville/Potomac MD… open to other areas.

We have exhausted Zillow.

Anyone have anything hitting the market soon?