Another experimental approach is to have the subjects perform some task before and after viewing the video, where their individual task performance varies according to the subject's emotional state: comfort vs discomfort, relaxation vs agitation, calm vs upset. Ideally, the task is simple and not something that involves thought or skill. In this setup, your goal is to have subjects reveal their emotional state through their performance in this task, and not through any direct revelation or awareness.
The other advantage of this approach is that you aren't signaling to the subjects that you are testing for their emotional state.
An example of such a task is interpreting the emotional state of another person's facial expression. For example, the task might be to evaluate a series of photos with ambiguous facial expressions and put them into two categories: "Trusted" vs. "Not trusted". (You can put this in a simple setting: "Imagine that you are in the library. You need to step outside to take a call. Would you trust this person to watch your valuables (computers, books) for a few minutes?")
Though I don't have a reference for you, I believe I read some research on this that found that subjects were more likely to interpret ambiguous facial expressions positively if the subject was in a positive emotional state.
There are even simpler tasks that might be good candidates. You could probably find a list of candidates by reading the experimental research literature.
Again, all you care about in this experiment is the difference between each individual's performance when they are upset vs. at ease.
You can test the validity of this experimental method independently. Recruit some subjects for a randomized control trial. Instead of showing them the video, you will directly irritate them (stimulus group) or not (control group). Your stimulus could be 15 second audio recording of fingernails scraping over a blackboard. For the control, you could use 15 second audio recording of a sin wave tone (around 400 hz). Your performance on pre- and post-tasks should be very different for the stimulus group and show no change in the control group. (For best results, you should randomly assign subjects to groups, and even you should not know who is in what group -- "double blind".)
The besides validity testing, this experiment will give you a standard of comparison to evaluate just how irritating or upsetting your videos are, compared to fingernails scraping over a blackboard.
EDIT: If you know the TV show "Mythbusters", they have done experiments involving sleep deprivation and also drinking where the subjects had to perform the same task before and after the stimulus. As I recall, the "beer goggles" episode involved a task of scoring faces for attractiveness before and after drinking heavily.
EDIT: The other answers advise you to measure physiological phenomena through direct sensors. Even if you can get the sensors and data recorders, analyzing the data can be more complicated than it sounds. It's not so easy to aggregate those data into a reliable overall assessment of emotional state along a single dimension: at ease vs upset. Inferring emotional state from physiological variables is a whole research field in itself!