Where was the Big Bang?
It is all in the dimensions!
As the balloon is inflated, the dots get further apart, just like the galaxies in the Universe - it doesn't matter where you are on the balloon, it will always look about the same. However, the Universe of the balloon is just the surface - it is two-dimensional. Therefore, the centre of the "ballooniverse" is not part of it - it is outside.
Although our Universe is three-dimensional, the "centre" is similarly meaningless - you would need to be able to "see" in more than three-dimensions to be able to find it!
Do not worry if this is a bit confusing. We have three-dimensional minds, and so imagining things which are more complicated than that is almost impossible.
Scientists use mathematics to describe the Universe. They then use these mathematical descriptions to make predictions of what they should be able to see in the real Universe.
These predictions are then tested by making observations with telescopes and so on. Only if the tests are all passed is the description used, and even then scientists will keep testing and testing until they come up with something better!