What has just happened?

Your friend had incorrectly computed the implications of his prior to the problem in question. On your prompting he re-ran the computation, and got the right answer (or at least a different answer) this time.

Perfect Bayesians are normally assumed to be logically omniscient, so this just wouldn't happen to them in the first place.

What do we say when we see the dart land at a specific point on the line?

In order to specify a point on the line you need an infinite amount of evidence, which is sufficient to counteract the infinitesimal prior. (The dart won't hit a rational number or anything else that has a finite exact description.)

Or if you only have a finite precision observation, then you have only narrowed the dart's position to some finite interval, and each point in that interval still has probability 0.