Are there celestial bodies that are about as far away as the Sun?
Asteroids with an orbit of between 1.8 and 2 Au (So their closest approach is is about the same distance to the Earth as the Earth is to the sun) are called Hungaria group asteroids, named after 434 Hungaria. 

Hungaria itself has an orbit of 1.94 AU, so would be a good example of body. Such asteroids are rather rare. There is a gap in the asteroid belt caused by Jupiter at 2.06 AU, and tighter orbits tend to be disrupted by Mars.

Of course, at any given time, the actual distance to Hungaria will vary between about 1 and 3 AU