You have to define 'Information' before you can measure its value.
Is information something that exists only in our minds? Are 0's and 1's actual information before we process it?
Are there any information stored (hidden variables) in particles before we measure it? Quantum Mechanics suggests that there aren't - and that particles only attain 'real' properties when measured or otherwise observed.
If you are referring to entropy the question is of quite a different matter. And I would believe that the number might be measurable unless we assume that the universe is infinite, which of course would imply that entropy also be infinite. But in a closed system, such as you reefer to, I think that entropy might be measurable.