Is it possible to track an handheld shot (there is already an answer on here about tripod shots: Tracking long panning shots) that doesn't have any visible points in common between the first and the last frame? As an example, imagine a shot of someone driving in a car for thirty minutes. Obviously everything from the first couple of seconds won't be there anymore at minute 15.
Yes, it is possible to track and reconstruct shots where the beginning and end of the movement have no common elements to track.
The important concept here is that there has to be a section of the shot that contains enough (and accurate) tracking points (at least 8 of them), and good parallax information to be able to solve that section. Blender can then reconstruct beyond that point, even on places where there are no "bundled" tracks. Note that you should track a whole bunch of elements as they move through the shot, from the moment they enter frame until they exit.
The trick to pull this off is choosing the section of the shot to be used. That is what the Keyframe A and Keframe B are for. In most cases you'll get good results if you set blender to determine the most usable section of the shot by enabling keyframe
If a tracked element leaves the frame but comes back later in to the shot, it is very important that you track both sections and then join the trackers so that blender understands that is dealing with the same element and not something new.
For more tips on Motion Tracking please read through the following link:
If you're looking for a more automated way to track long shots with a constantly evolving environment, I recommend having a look at my answer on this similar question: https://blender.stackexchange.com/a/209119/16265