Do asteroids come close to earth?
Not usually, but sometimes; sometimes the Earth even collides with small asteroids! Perhaps you wouldn't call a rock 1 meter in diameter an "asteroid", but a rock about that size collided with Illinois on April 15, 2010, startling millions of earthlings with a brilliant explosion in the sky.
You can watch the video of it at spaceweather.com, at the link below.
Larger asteroids sometimes come quite close to the Earth; there was one about 3 months ago that passed in between the Earth and the Moon. And on Friday, April 13, 2029, the asteroid Apophis will come within 20,000 miles of the Earth - but it will miss.
Most asteroids are in the "asteroid belt", a roughly toroidal (meaning "doughnut-shaped") region of space between Mars and Jupiter.