I have the spell Find Familiar which partially reads:
You gain the service of a familiar, a spirit that takes an animal form you choose.
Your familiar acts independently of you, but it always obeys your commands.
This question suggests that your familiar will try to follow your commands to the best of its ability, but if the command is too complex for the simple mind of the animal to understand (DM decision based on their Intelligence) they can't follow through on it.
Would it be reasonable for a summoned Owl familiar to be on guard duty while my PC is asleep? Circling around the sleeping place or just sitting on a tree nearby and trying to wake up my PC when something out of the ordinary happens?
I could imagine this command to be worded like this:
Have a close look on our surroundings and if you see sudden movement or signs of other creatures like fire or sounds try to wake me up.
Things to consider:
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The familiar obviously needs to sleep as well (as suggested by this question), this would happen the day after when my PC is awake by sending it to its pocket dimension for a reasonable time (8 hours probably).
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The familiar is probably not able to determine what exactly is out of the ordinary so it might happen that it wakes the PC up for things that after a short investigation turn out to be nothing of big interest. Would these interruptions be too much for the rest to still be considered a long rest following the rules?
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The familiar and the PC have been travelling together for a long time. The familiar is not really a wild animal anymore but dependent on the PC and has probably been trained to a certain degree like a pet. It will be rewarded for keeping guard and waking up the PC for positive reinforcement.
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The owl has an intelligence of 2 which is not exactly to be considered overly intelligent. Comparing to real life this fits the fact that owls are not to be considered clever, even among birds. However there are trained owls that can at least do basic things like flying to certain places and returning in a given time-window.
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As nwp pointed out the familiar is not actually an animal but a celestial/fey/fiend instead of a beast. Could this impact the intelligence? The spell description states the familiar has "the Statistics of the chosen form" which would include Intelligence.
Best Answer
Simple answer: yes
Before you go to sleep, you can communicate clearly enough to leave the owl instructions to be alert and then do something if it detects something. (Wake me up, hoot as loudly as possible, etc). The text you quoted states that it will obey your instructions. Since it can act independently of you, you can be asleep and it can be awake.
But there's a catch. The owl is not guaranteed to detect danger.
From the Basic Rules, DM, p. 39 (owl's stat block, excerpted)
While it's likely that the owl will detect danger that can be seen or heard, since it has darkvision and advantage on hearing/sight Wisdom(Perception) checks, a monster or NPC approaching by using Stealth may foil the owl's efforts at detection.
The owl may or may not detect danger, depending on whether or not the stealth check is successful as compared to the owl's perception check. If the stealth is more successful, then having the owl on watch won't result in an alert. This is the same risk as having a PC on watch: they may still get snuck up on if the approaching creature's Stealth score beats the PC's Perception score.
Experience
In our first campaign, my brother's wizard used an owl familiar that he assigned to help keep watch while he was trancing/sleeping while another character (like one of the 3 humans) was on watch. On a few occasions this helped alert us to approaching danger.