Age of the Universe
Now I will not use the geological evidence for an old Earth. There are a couple of reasons for doing this: first, though I have an understanding of how geologists have worked out the age of the universe my training is in astronomy, so I will only use astrophysical evidence for an old universe. Second, the radiometric method used by geologists to date rocks is panned by creationists so they'll have to come up with other reasons that the age of the universe is only around 10 kyr (k=1000) as they seem to claim (Christnet, Genweb). The point being that radiometric dating isn't the only way we have determined that the universe is on the order of 13 Gyr (G=1 000 000 000) old, not a few thousands year old.
White Dwarfs
Now stars around the size of our Sun end their days as white dwarfs (Iben 1991). These white dwarfs are basically the exposed cores of the former star. Now when in the middle of the star, these cores had temperatures in the millions of degrees. Now exposed to the universe after the outer layers of the star have blown off into space, it will take time for these cores to cool. In fact the amount of time for these to cool is longer than the age of the universe (at 13 Gyr old). That being said, these objects cool at roughly the same rate, meaning that they can be used to gauge the age of the universe. Using only white dwarf data, the age of the universe is at least 10 Gyr old (Winget et al 1987). This determination is completely independent of any other age determination thus is "uncontaminated" by the radiometric dating technique and it comes up with an age of the universe in the billions of years.
Cosmic Microwave Background
This is not the only determination of the age of the universe in astronomy however. Data from the Wilson Microwave Anisotropy Probe (WMAP) has shown that the age of the universe is around 13.7 Gyr (Komatsu et al 2009). It determined this by measuring the echo of the Big Bang, the cosmic microwave background (Dicke et al 1965). Again a universe the age of which is in the billions of years.
Stellar Evolution
One can also look at stellar evolution, that is the life cycle of stars to see that the universe is much older than the order of 10 kyr. Stars are big. The Sun, a small star has a mass of 1.989e30 kg, thats basically a 2 followed by 30 zeros. Stars range in mass from smaller than half the mass of the Sun to up to 120 times the mass of the Sun. Stellar mass is important as the more massive a star is, the brighter it is and hence the faster it burns up the nuclear fuel in its core. So basically, the larger a star is, the hotter it is and the hotter a star is, the faster it consumes its nuclear fuel (Iben 1967). This means that the larger a star is, the shorter the time it survives. Now it can be calculated how long a star can remain in a state of burning nuclear fuel in its core as this is a function of how much fuel is available which in turn is a function of the star's mass. To this end, a star that is around 15 times the mass of the Sun will only live about 10 million years (10 Myr) (Iben 1967). This means that for a universe that creationists claim is only 10 kyr old, we should not have any evidence of stars of this size having yet died, as the universe would not be old enough for this to happen. Unfortunately for the creationists, there is evidence of stars of this size having died. This evidence exists since these stars not only live fast and die young, they die spectacularly in supernovae explosions. Several of these explosions have been observed over the centuries, and in 1987 one was observed in a companion galaxy to our own Galaxy. Since supernovae 1987A was so close, we were able to determine the progenitor star, that is the star that blew up. This supports the fact that supernovae are caused by large stars at the end of their life exploding. Further, in the location of historical supernovae, such as the one observed in 1054 by Chinese astronomers, the remnants of that explosion are still visible as a nebula of gas and dust today known as the Crab Nebula. This puts at a minimum the age of the universe at 10 million years, 1000 time longer than what creationism proposes. Further smaller stars like the Sun live about 10 billion years before, as stated above, becoming white dwarfs. Since white dwarfs have been observed (as well as the stage between star and white dwarf known as a planetary nebula) the universe has to be at least 10 billion years old, which is supported by the white dwarf data. So the lives of the stars themselves show that the universe is a very old place.
As a sideline to this, stellar evolution shows that stars like the Sun last on what is known as the main sequence (the curvy band through the middle of the graph below) for about 10 billion years. The stars aren't static while on this band, they start at the bottom and move roughly straight up, which is why the band has a width to it. By seeing where a star is in the band one can, as a first order approximation, determine how far the star is through its lifespan. The Sun is about the middle of the band, thus meaning it has used up about 50% of its 10 billion years, making the Sun roughly 5 billion years old. This is in general agreement with radiometric dating of the Earth at roughly 4.5 billion years old.
Hipparcos colour-magnitude diagram (Hipweb)
Light Travel Time
This leaves the patently obvious, that is light travels at finite speed and space is big. Space is so big that any object further than 10 thousand light years (a light year is the distance light travels in a year) would be invisible to us here on Earth since its light won't have had time to reach us yet. So the Andromeda galaxy, visible to the unaided eye as a faint fuzzy patch in a dark northern sky would not be visible as it is around 2 million light years away. This means that if you were to go out and see it tonight, the photons entering your eye left the galaxy 2 millions years ago. Way more than the 10 kyr of creationism. And objects have been viewed to the edge of the observable universe over 13 billion light years away, meaning that light has been travelling for that long.
An Old Universe
So there you have it, four different ways of measuring the age of the universe, or at least constraining the lower limit of that age. None are dependent on radiometric dating and all have the universe in the billions of years old. One can even be said to support the findings of radiometric dating for the age of the Earth itself. So the evidence shows that a young universe as proposed by the creationist/intelligent design crowd is pure and utter bunk. Of course good luck trying to convince them of this.
References:
Christnet, AGE OF THE EARTH - Is the Bible clear about the age of the earth and universe? - ChristianAnswers.Net, http://www.christiananswers.net/q-eden/edn-c002.html (Accessed 21 Jul 2010)
Dicke, R.H., Peebles, P.J.E., Roll, P.G., Wilkinson, D.T., 1965, ApJ 142, 414
Genweb, The World: Born in 4004 BC? - Answers in Genesis, http://www.answersingenesis.org/articles/am/v1/n1/world-born-4004-bc (Accessed 21 Jul 2010)
Hipweb, http://www.rssd.esa.int/SA-general/Projects/Hipparcos/images/f3_5_005.pdf (Accessed 21 Jul 2010)
Iben, I., 1967, ARA&A 5, 571
Iben, I., 1991, ApJS 76, 55
Komatsu, E., et al, 2009, ApJS 180, 330
Winget, D.E., Hansen, C.J., Liebert, J., Van Horn, H.M., Fontaine, G., Nather, R.E., Kepler, O, Lamb, D.Q., 1987, ApJ 315, L77