Varuna is one of the few formally named Kuiper belt objects because it's relatively large -- in fact, for a time, it was the second largest known object in the Kuiper belt, after Pluto. It's always difficult to know exactly how big such a distant object is, though. You can estimate the size by making an assumption about the albedo, which is a measurement of how much of the Sun's light the body reflects. But albedos of objects in the solar system can vary widely, from the incredibly frosty white surface of Enceladus to the blacker-than-coal exterior of comet nuclei. Kuiper belt objects' albedos probably vary within a narrower range than that, but still, there's well over a factor of two range, which leads to pretty large uncertainties in the sizes of these things.
Varuna's diameter is better constrained than most at somewhere between 755 and 1,029 kilometers (usually reported as around 900 kilometers, about the diameter of the largest asteroid, Ceres) and a pretty dark albedo of around 7%, based upon simultaneous observations performed with optical and infrared telescopes by David Jewitt and his coworkers in 2000. It's all explained in this paper in the journal Nature (which is unfortunately available by subscription only, but you can at least read the abstract and the assorted news articles that came out at the time this paper was published).
Well, there's an opportunity coming up to get a much better estimate of the size of Varuna; it's about to pass across a faint star as seen from Earth, an event called an occultation. If several astronomers watch the occultation and share their information on the exact times that the star winks out and then reappears, astronomers can calculate the diameter of Varuna. With just one observatory you'd just get a lower limit, because you wouldn't know whether the star passed exactly behind Varuna's center (in fact, it probably didn't), but if you get enough observatories you can figure out what parts of Varuna passed across the star as seen from different places and get quite a good estimate of the diameter.
So Dave Herald has issued a call for astronomers, professional and amateur, to observe the occultation. Simulations suggest it'll only be visible from South America and the southern parts of Africa -- Brazil and Namibia would be the best observing sites. Current predictions suggest the occultation will take place at around 2:10 UTC, in the early hours of December 7. Here's all the details -- I wish clear skies to those of you who attempt observations!