If a bridge was built about 3 feet off the ground around the circumference of the Earth, and then all the supports holding it up were removed, what would happen to the bridge?

It would buckle under its own weight. Whether it broke into pieces would depend on the result of the buckling as it is very near the surface of the earth.

But, in broad terms how big would it need to be to support itself.

The circumference of the earth is between 24,859 – 24,901 miles. Let’s call it 24,900 miles and assume a path has been prepared so no issues with mountains or such. I believe that such a structure would act like a simply supported beam of half the circumference so 12,450 miles. Again assuming a steel static shape section like a box beam or wide flange we would have a 20:1 size for a nominal bridge load so it would need to be about 622 miles tall to support the weight and about 60 miles wide to support nominal torsion. So that covers half of low earth orbit.

We would need to place this thing higher than 3 feet off the ground as normal deflection of an object that big would be more than 3 feet as would the likely shift from the moon.

But in theory, you need a bridge roughly 620 miles high by 60 miles to hold a stable position. The further out it gets the bigger it gets.

Keep in mind that these are very rough design calls and the actual should be 10% or so smaller once worked out.