It's based on SMS [0], the standard that allows telephones to receive text messages. The message size in SMS is limited to 1120 bits, which works out to 140 8-bit characters (or 160 7-bit characters.) One of twitter's design goals was to be usable from the broadest set of mobile devices possible.
SMS/text message length, minus the site identifier; lets people use the service via cellphone. As for why text messages are restricted to that length, word is that the guy who developed the first system felt 160 characters was plenty for most communication needs, based on the average length of a random sample of sentences and considering the tedium of having to enter the message via the 12 buttons of a then-typical telephone keypad.
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[ 3.2 ms ] story [ 19.8 ms ] thread[0] http://en.wikipedia.org/wiki/SMS
So ultimately, it's arbitrary but practical.