More specifically, would I be able to keep watch through the eyes of a familiar, and still get credit for a short rest? Or a long rest?
Long Rest, certainly not.
Most races have to spend 6 of the 8 hours of a Long Rest asleep (Elves and Half-Elves only Trance for 4 of the 8 hours). Even the ones that don't sleep must spend that time 'in an inactive state.'
A long rest is a period of extended downtime, at least 8 hours long, during which a character sleeps for at least 6 hours and performs no more than 2 hours of light activity, such as reading, talking, eating, or standing watch. If the rest is interrupted by a period of strenuous activity — at least 1 hour of walking, fighting, casting spells, or similar adventuring activity — the characters must begin the rest again to gain any benefit from it.
From Chapter 8, PHB
The 'see through their senses' ability of a familiar has to be constantly renewed as an Action
While your familiar is within 100 feet of you, you can communicate with it telepathically. Additionally, as an action, you can see through your familiar's eyes and hear what it hears until the start of your next turn, gaining the benefits of any special senses that the familiar has. During this time, you are deaf and blind with regard to your own senses.
Emphasis Mine from the Find Familiar Spell
If you are taking an Action every 6 seconds to keep renewing your sensory link with your Familiar, you are obviously not asleep or inactive.
A Short Rest is a little less clear.
A Short Rest's rules for 'resting' are actually more strict than those of a Long Rest. A Long Rest allows up to 2 hours of reading, talking, eating, and standing watch. A Short rest does NOT list 'Standing Watch' as a valid option for what you're doing while on a Short Rest.
A short rest is a period of downtime, at least 1 hour long, during which a character does nothing more strenuous than eating, drinking, reading, and tending to wounds.
So, does using an Action every 6 seconds to renew your link to your familiar's senses count as 'More strenuous than eating, drinking, reading, [or] tending to wounds?'
There is nothing definitive here from the developers or in the rules...so the interpretation of this would be left to your DM.