The word sak שק in Hebrew and "sack" in English have the same meaning, and this is no coincidence. The English word derives from the Hebrew:
"large bag," O.E. sacc (W.Saxon), sec (Mercian), sæc (Old Kentish) "large cloth bag," also "sackcloth," from P.Gmc. *sakkiz (cf. M.Du. sak, O.H.G. sac, O.N. sekkr, but Goth. sakkus probably is directly from Gk.), an early borrowing from L. saccus (cf. O.Fr. sac, Sp. saco, It. sacco), from Gk. sakkos, from Semitic (cf. Heb. saq "sack"). The wide spread of the word is probably due to the story of Joseph.
The American Heritage Dictionary also shows the connection:
Common Semitic noun *saqq-, sack. sac, saccade, sachet, sack1, sack2, satchel; cul-de-sac, haversack, knapsack, rucksack, from Greek sakkos, coarse cloth, article made from coarse cloth, from a Semitic source akin to Hebrew saq, Akkadian saqqu, sack.
In the story of Yosef we find two uses of the words sak.
In Bereshit 37:34 we read וַיָּשֶׂם שַׂק בְּמָתְנָיו - that Yaakov placed sak - sackcloth - on his loins as a sign of mourning.
In Bereshit 42:25 Yosef gave orders וּלְהָשִׁיב כַּסְפֵּיהֶם אִישׁ אֶל-שַׂקּוֹ - to place the brothers money, each in his own sak - sack. (In this section we also find the word אמתחת amtachat - bag or pack. For the differences between sak and amtachat, see here.)
Which meaning came first - that of sak meaning "bag" or sak meaning "sackcloth"?
Both Klein and Kaddari give the meaning of "bag" first in their dictionaries, which would seem to indicate they believe it had the earlier meaning. But certainly mourners were not actually wearing bags (with the arms and legs cut out?).
So it would seem to me that there was an earlier meaning of this term. We can see this meaning by looking at a verse where it is not clear whether sak means sack or sackcloth - Vayikra 11:32 :
וְכֹל אֲשֶׁר-יִפֹּל-עָלָיו מֵהֶם בְּמֹתָם יִטְמָא, מִכָּל-כְּלִי-עֵץ אוֹ בֶגֶד אוֹ-עוֹר אוֹ שָׂק, כָּל-כְּלִי, אֲשֶׁר-יֵעָשֶׂה מְלָאכָה בָּהֶם
"And anything on which one of them falls when dead shall be unclean: be it any article of wood, or a cloth, or a skin or a sak - any such article that can be put to use..."
The JPS translates sak as "sack", but Levine corrects it to "sackcloth". Onkelos also seems to go in this direction. When sak means "sackcloth" as in the verse above, he translates it with the Aramaic שק, with the letter sin. But when it means sack, bag, he uses the homonym סק, with a samech. In this verse - he translates with שק.
Levine points out - I assume on the basis of Shabbat 64a - that there is a parallel between our verse in Vayikra and that in Bamidbar 31:20 :
וְכָל-בֶּגֶד וְכָל-כְּלִי-עוֹר וְכָל-מַעֲשֵׂה עִזִּים, וְכָל-כְּלִי-עֵץ--תִּתְחַטָּאוּ
"You shall also purify everything woven of cloth, every vessel of leather, everything made of goat's hair, and every vessel of wood".
Both verses have cloth, wood and leather, but here sak is replaced with things made of goats hair. To me this seems to be the original meaning of the word - sak meant either bags made of goat hair, or the uncomfortable garments of a mourner (or someone repenting or causing others to repent) made of goat hair (like the Christian cilice.) This site tells us that "at Qumran, Masada, and other caves in the Judean desert, articles made of wool, cotton, and goats’ hair were discovered; the latter was usually used for sacks".