The smallest unit in a segmentation is a single voxel. If you want to represent thinner structures than that then you need to oversample the segmentation as described here.
3D-printed model size will very accurately match the true size of the anatomy. Precision is limited by layer thickness and minimum wall thickness of your printer; and resolution (voxel size) of your segmentation.
You can measure volume of segments using Segment statistics module.
First of all, the image above is so low quality (bright areas are oversaturated, noise is overamplified) that your error is expected to be high. If it is not just an incorrect window/level (see how to adjust it here) then you may consider trying to obtain better quality images.
If you want to measure vessel diameter at only a few places then find those places, zoom in, and measure distances using Markups line or Annotation ruler tools. There is no need to segment the images, just look at the images and determine where the boundaries of the vessels are, based on your expert judgement. If you want to measure vessel lumen radius across the entire vessel tree then you can use SlicerVMTK extension’s Centerline extraction module (it requires segmentation of the vessel tree as input).