What are the possible measurement error barycenter of the Solar System in NASA data?
The barycenter cannot be directly observed. It is instead a computed quantity. The location of the barycenter would be easily determined if the universe was cartesian (it isn't), if we could simultaneously observe the locations of all of the objects in the solar system (we can't), and if we knew the masses of all of those objects (we don't).
In developing a solar system ephemeris, the modelers start with an a priori set of masses, positions, and velocities of the objects that comprise the solar system at some epoch time, and then propagate that backward and forward in time. The propagated states are compared to observations to yield error estimates. There are lots of incomplete and somewhat erroneous observations of these objects, scattered over time. These error estimates are used to suggest a better estimate of the state (mass, position, and velocity) at the epoch time that (ideally) reduces the accumulated weighted square error. The cycle repeats until no further improvements are seen.
Estimates of an object's mass (more specifically, it's gravitational parameter) improves significantly on the first visit by an artificial satellite to that object. The estimates improve even more when we send something into orbit about a solar system object.
For example, the Juno probe is about to reach Jupiter and go into orbit about it. Assuming all goes according to plan, this will inevitably result in much improved observations of Jupiter's mass, position, and velocity. Since Jupiter is by far the largest object in the solar system other than the Sun, these improved observations will yield a vastly improved model of the solar system.
Distance from barycenter is written with the precision up to 14 decimal places (in AU). Angles are written with precision to 4 decimal places (in degrees). What are the possible errors for angle and distance?
If you instead request vectors you will get sixteen places of accuracy, of which at least five are pure fiction because the Sun's gravitational parameter is "only" known to about eleven places of accuracy. The Sun constitutes almost 99.9% of the mass of the solar system. Add in the four giant planets and you get over 99.99% of the mass of the solar system. The uncertainties in those five objects' masses, positions, and velocities are the key factors that drive the uncertainty in the location of the barycenter.