In Hebrew, the word seret סרט means "ribbon". Let's take a look at the history of the word, and some possible relatives.
As Elon Gilad writes here, the word first appears in the Mishna and meant "strip (of fabric)". In 1892, Ben Yehuda revived seret to mean "ribbon", and in the 1920s, it came to mean "film" and "movie", due to the ribbon-like appearance of the film strip. Klein says that sirton סרטון actually means "film strip", but today sirton is used for a short film, particularly a video clip (like the ones shared on social media).
Gilad says that seret may "derive from the Greek word sirtis, which means “bolt” (as in locking the door)." This is similar to the suggestion in Ben Yehuda's dictionary that it derives from the Greek syrtēs, meaning "rope", however the dictionary ultimately rejects this idea as unlikely.
Klein does not give an etymology for seret (other than saying that it is cognate with the Arabic sharit). However, he does tell us that there is another similar looking Hebrew word that derives from the Greek syrtis: sirton שרטון (this time with the letter sin, not samech). Here is his entry for this sirton:
Post-Biblical Hebrew - "sandbank". [Borrowed from Greek sytris (=quicksand), from syrein (=to trail, drag, sweep away), which is related to sairein (=to sweep, clean)]
Note that both "bolt" and "rope" are things that are pulled or dragged.
In modern Hebrew you'll often hear the expression "ala al sirton" עלה על שרטון - meaning metaphorically "to get stuck, run aground", like a boat on a sandbank. (Interesting that the Greek word referred to something you could get stuck in by sinking, whereas the Hebrew was the opposite - you got stuck higher up than you wanted to be.)
Syrtis today refers to two sandy gulfs in North Africa, called so either due to sandbanks or quicksand. Klein, in his CEDEL, writes that the English word "swerve" is cognate (both originating from the same Indo-European root, *swerbh), also having a similar sense to "trail, drag, sweep away."
However, there are other theories regarding the origin of seret. Stahl connects it to the root סרט, or in the Biblical form שרט, both meaning "scratch, scrape." The Arabic cognate also means "to tear", or "to rend one's garments in mourning." He writes that the Arabs would tear leaves and fibers from palm and other trees to make ropes and strips - and this is the origin of the word seret.
From this root we get the verb שרטט, "draw, rule, mark lines, sketch" and sirtut שרטוט is "drafting". Another related word is sartan סרטן. It originally meant crab (a scratching animal), which in the zodiac is the sign "Cancer", and as in English later came to mean the disease cancer as well. The connection between the crab and the disease isn't clear - perhaps the hard tumor is similar to a crab shell, or maybe the enlarged veins of a cancerous tumor resembled the legs of a crab.
To go back to our original word, in modern Hebrew slang, seret means an exceptional experience, often a negative one. Eizeh seret איזה סרט - "I can't believe what just happened to me." On the other hand, mehaseratim מהסרטים ("out of the movies") indicates something unusually excellent. I hope this post falls into the later category...