Can you use an directional Antenna to sniff packets at a very long distance?
In theory yes, if your antenna is big enough. In practice there are limits. Consider the Voyager spacecraft. It's currenctly some 20 billion km (2.0 x 10^10 km) away from earth and has a 20 Watt radio transmitter (that's about 200 times your average 100mW WiFi signal). On Earth we can still pick up the signal but it requires a 38 meter dish antenna (the Voyager also has a dish antenna, making the signal highly directional).

So if you want to go long-distance, you need at least a dish antenna. See the picture of a Cantenna on Wikipedia for something that looks like just what you need.

The same logic applies to your WiFi dongle; the only thing that determines the maximum receive distance is the power output of the transmitter and the sensitivity of the antenna and receiver. There is no limit inherent to 802.11n packets that says "stop after 250 meter"; radio waves continue infinitely far. The 250 meter limit is just the practical limit where current electronics are still able to transmit and receive reliably (Well, reliably.... perhaps in the middle of the Sahara with no other sources nearby. In a crowded urban environment you may not be able to have a reliable reception after 10 meters. But that's an entirely different question).