Can we determine the date just from sky observations? Assuming everyone "freezes" for several thousand years. Then everyone unfreezes at the same time. Could we tell how much time passed based on sky observations without fancy telescopes?
(Just started watching Dr Stone and I had this on my mind)
I'm guessing we could tell something based on how less sun does the moon cover compared to now during an eclipse. Am I on the right direction? Can we tell that with basic tools that don't need very precise cutting?
 A: A simple protractor, a clear view of the night sky, many years (at least a few decsdes) of careful observation, and some scientific reasoning is enough to build formulas, charts, or models that can tell you  approximate date and time anywhere on the earth.
A good resource for this would be links found by googling "sky navigation ship".
A: Easily observable features (i.e. without the need for any
telescopes) are the north pole and south pole of the sky.
These are the points in the sky around which all stars seem
to rotate during their nightly course.
Currently the north pole of the sky is near the star Polaris
($\alpha$ Ursae Minoris).
But due to the precession of the equinoxes this is changing
slowly through the millennia.
The north pole of the sky circles around the ecliptic pole in 23.5°
distance once in 25772 years.
The timeline of the north pole from 10000 BC to 14000 AD looks like this (notice the "Polaris" star near the "+2000" mark):

image from Precession of the equinoxes - Changing pole stars

Another long-term effect changing the apparent positions of the stars
in the sky is their proper motion.
The stars move slowly (mostly with speeds of a few milliarcsecond/year)
across the sky relative to the background of the more distant stars. 
Here is how the constellation Ursa Major (i.e. the Great Bear)
looks now and 100000 years from now.

image from astropixie: ursa major
A: *

*The earth's polar direction do change.

*The solar sun is traveling in the space, so does the motion of the galaxy.
However, you got a measurement issue: you want a number without precision measurement. Yet, by your means of determining the date, it does require precise measurement. 
But I guess, one may use relative position of the planet to determine the year.
However, if you can't even make the precise measurement to establish a first reference, it's highly impossible for you to measure the date in a digital form.
