#1. Experimentally, I'd say no. There's no such thing as an absolutely straight line in the real world. Therefore, you couldn't measure length or area exactly, because you'd have to assume straight lines somewhere, which don't exist. Any technique you could think of to measure volume would have inaccuracies.
Theoretically, you also have problems: just think of length. As you use smaller and smaller ruler lengths (as in, microscopic), you can prove that the length goes to infinity (you're moving back and forth more and more). If you're going around molecules, and you run into quantum mechanics, you now have the additional difficulty of defining length in the first place.
#2: Again, I would say no, for reasons I've outlined above.
The branches of mathematics to which this question belongs would include fractals, quantum mechanics, and experimental science.