Yes and no. Disclaimer: I've been obsessed with bow making for many years so any question you ask I will probably end up telling you much more then want to know.   So short answer, yes it will make a bowstring.

It will be big and clunky and have a low mass per breaking strength, but with a crossbow that may not matter as much. Never made one so I can't tell you. The big problem with big, clunky strings in regular bows (and stretchy strings like artificial sinew which is just waxed nylon) is they slow down your bows potential ALOT. So if you make a crossbow that is already fairly weak a jute string might slow it down to something slightly better then a wet noodle, where as one made of a fast flight material might make it a decent weapon. A BETTER natural fiber twisted into a well-made string will give you something a lot better then jute, though not up to par with synthetic fibers.

Jute is considered one of the weakest natural fibers for bowstring making. Weak=need more=more mass=saps energy=slower bow. But again it will work, and for a crossbow it probably won't make as much of a difference. As far as readily commercially available fibers that you could go to your local store and get you could try the thinnest hemp available EDIT: Obviously twisted into a cord of sufficient strength  . (commonly called Jewelry hemp) . If artificial fibers don't offend your sensibilities just by some spiderwire, braided kind. It's a dark green and looks like thread. 150lb breaking strength.

Last but not least, if you give me the finished size you will need for the string I would be happy to make you one free of charge, as string making is enjoyable to me.  

Well...there's a mouthful for you.