This picture of Eros, the first of an asteroid taken from an orbiting spacecraft, is a mosaic of four images obtained by NASA's NEAR mission immediately after the spacecraft's insertion into orbit.
Question: How close has an asteroid come within the Earth’s gravitational field without a collision occurring?
Answer: Asteroid collisions with Earth are a very rare occurrence with the last impact called the Tunguska event occurring on June 30 1908 in what is now Krasnoyarsk Krai, Russia; this was the largest impact event over land in Earth's recent history with estimates of the energy of the blast range being approximately 10–15 megatons of TNT (1,000 times as powerful as the atomic bomb dropped on Hiroshima).
It is estimated that collisions that produce explosions of comparable scale only occur every 2,000-3,000 years, for some perspective, asteroids with a diameter of one kilometer hit the Earth an average of once every 500,000 years and large collisions with five kilometer objects occurs approximately once every ten million years.
Small objects called meteor’s frequently collide with the Earth, these are formed when an object (often a broken piece of an asteroid or comet) falls into Earth’s atmosphere. As the Earth has an atmosphere its surface is protected from collisions with meteors up to a moderate size mostly evaporating upon entry, these can be seen as "shooting stars".
You can read more about monitoring comets and asteroids orbiting near the earth in NASA’s Near-Earth Objects Program.
An asteroid near miss is measured by the Torino scale which is a method for categorizing the impact hazard associated with near Earth objects such as asteroids and comets. The scale scores from 0 to 10 with 10 being a definite impact that will devastate most of Earth. Calculations undertaken in December 2004 of the asteroid ‘Apophis’ with a size of 270 meters and a mass of 2.1 x 1010 kg, was predicted to have a 2.7% chance of hitting the Earth in the year 2029! Additional observations later ruled out any possibility of an impact in 2029, however, the asteroid is expected to make a record-setting, (but harmless) close approach to Earth on Friday, April 13, 2029, when it comes no closer than 18,300 miles above Earth's surface.