How did scientists measure the size of the universe? How accurate is that figure? Can we calculate the volume of the universe? What is the total mass of the universe?

The approximate size of the universe is measured by many means depending over what scale one is measuring...parallax, type 1a Supernova, Ceipheid Variables, cosmological redshift are the more common methods. Errors and conflicting answers do occur but comparing different measuring means gives us a reasonable answer.

The observable Universe is what is being measured, which is basically a function of dark energy and time since the Big Bang. So about 90 billion light years across.

It's quite interesting, actually. There's a "cosmic distance ladder" that allows us to gradually work out the distances to more and more distance objects. Different methods are used for different distance ranges, but the different methods overlap to some extent, so we can check one against another. One important concept is the "standard candle", which is an object whose intrinsic brightness is well known. By observing the apparently brightness and comparing to the intrinsic brightness, we can work out the distance to the object. We determine the distance to the most distant objects (usually galaxies) by measuring the red shift in the light they emit and using Hubble's law. More and more accurate every year. Since the size of the universe is related to its age, we're also pinning down the age of the universe to greater and greater accuracy. Of the observable universe, yes. Just use the volume of a sphere of the right radius. About $10^{52}$ kg. That's equivalent to the mass of about 25 billion galaxies of the same mass as the Milky Way.