Okay… I might not be able to make this sound right… but i’ll try…
When we look up in the sky… at the sun… based on our distance from the sun and its size… is it to scale?
For example… if we round the sun’s size to 1 million miles wide and our distance from the sun to 100 million miles… is the sun in the sky directly proportionate to something that wide and that far away? or if the atmosphere distorts it, how much?
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[/codeOops meant to press preview, ignore my attempt at an ascii diagram.
The formula for angular size IIRC is: a = 2tan[sup]-1/sup
where a is the angular size, r the radius of the observed object and d the distance to the observed object. Objects with the same angular size will appear the same size in the sky.
hehe, it looks like MC drew a really long. . . . . hehe, ya know <blushes>
Well the impact of the atmosphere is going to be small-ish depending on where you live and what you’re looking at. The index of refraction of air at STP is 1.00029 while a vacuum’s is 1.0. The change in size is likely small enough as to not really matter. Remember that the atmosphere thins the higher you go. 5.5 km straight up half the atmosphere is below you.
However the further away from right above you the object is the greater the impact of the atmosphere because the incident angle of light moves further from 90 degrees.
This might help. atmospheric refraction
Long ago, I read that atmospheric lensing of the setting sun is only enough to displace its visual position by half a degree from the true direction to the sun. The discrepancy between the actual and apparent position of the sun will be less at other times of day, when the geometry makes for a weaker lens between sun and the observer.
Like the moon, the sun stays the same visual size whether it’s high in the sky, or near the horizon. This fact precludes any significant magnification effects from atmospheric lensing.
Of course, the sun is only half a degree in angular diameter.
It’s big. Bigger than the biggest thing ever and then some. Much bigger than that, in fact, really amazingly immense, a totally stunning size, real ‘wow, that’s big,’ time. 
Watch the BBC version of Hitchhiker’s Guide to the Galaxy.
Try taking a look at http://www.spaceship-earth.org/astro1.htm
The idea is to imagine that the earth was 1mm in diameter. Then the Sun would be 12m from it, and the solar system would be about a kilometer wide. The galaxy would be 73,000,000 km wide!
You would have to do a similar excersize to grasp the size of the universe relative to the galaxy.
IMHO the answer is “You can’t grasp it. Sorry.”
smootman, I declare you a Victim of your Own Thread Title. People will answer the question they expected you to ask based on your title, and not the question you actually asked. 
Having said that, this whole “size of the Universe” thing did get me thinking. An object which is 1 km in size, at a distance of 1 million km, will in fact appear the same size as an object 10 km in size at a distance of 10 million km, which is what you’re asking in your OP. However, an object 10,000 km in size, at a distance of 10,000 million km, will not necessarily appear this same size, because on scales that large, the universe may be curved.
This site tries to give some perspective.
You know, I said km, but I was thinking light-years. The universe may be curved on the scale of billions of light-years.
The universe looks to be flat, but yes if it were curved objects on the absolute outer reaches of the visible universe would be gravitationally lensed.