A Story of the Machine that Spits Out Beans Tale of a Bean Spitting Machine [There is nothing wrong with your title grammatically but too long and not interesting enough.  Titles are like advertisments, short, crisp and intriguing :) ]

There was a village. A where a family invented made up a machine which could can produce beans when fed with something else. always put out different bean while fed with different stuff. Since then, with the this machine, the village to be became more prosperous day by with each passing day.

One day, a child put into with the bean fed the machine with a bean.  which was put out by The machine itself, then, it was just running whirled and twirled incessantly but nothing came out of it. couldn’t put out any bean. Finally, the people had to take it [It is obvious]  machine had to be taken apart.  to pick After the culprit bean had been removed, out and then return to normalcy returned to the machine.

After that, The elders decided to set the a rule up: Never put the beans back into feed the machine with beans! It was ended, for the people who utilize the machine. Can't figure this out. 

The family who made the machine heard the incident and was about this , however, to be very upset. They tried Painstakingly, wanted to make a more [perfect means no flaws, so cannot have "more" to describe it.  Another word is "favourite" (best).  Cannot be "more favourite"] they want to perfect the machine which when fed with the same input, will not fall into a tizzy.  couldn’t be in trouble by feeding into the beans spitted itself meanwhile kept that, just like the original, what bean out as what stuff in. For thisthat, many tried a lot of smartest guys till they vomitted blood to find a solution.  worked to vomited blood, but never got nary a satisfactory outcome. [What you wrote is a Chinese expression.  It gives a fresh flavour to the writing.  However, you must not use too many as it would be too "gaudy".  In standard English, we can simply write like this, "Many tried but to no avail"]

The machine, what the people in the village couldn’t live without, was called as formal logic. The family made the machine, known as logician. Don't see how this line can be the punch of the story and how it is related to the story. If you write this in Chinese, hopefully my Chinese is good enough to understand it.

On the whole, a nice story and quite well written.  Not too many grammatical mistakes.  However, there are places where the description does not flow well.  Storytelling is an artform to entrall the reader.  As you can see from my style, there are additional and unnecessary words to add 色彩 to the tale.

Now, with the original source,  here's how I would translate it as...

有个村子，一个聪明的家族造了一台神器的机器，喂它不同的东西，就吐出不同的豆子。有了它之后，这个村子就日益繁荣起来。In a village, there was a smart family who made a marvellous machine.  Feed it with different thing and it will spit out beans.  As time passes, this village began to prosper.

某天，一个孩子把机器吐出的豆子喂回给它，机器不停地运转，却怎么也吐不出豆子。最后，只能将机器拆开，取出那粒豆。从此，长老们立下一条规矩：不准将这机器吐出的豆子喂给它自己。

One day, a child threw back in the beans the machine had spat out.  The machine incessantly twirled and whirled but nothing came out.  Only when the machine was taken apart and the offending beans were removed.  [The original was a bit inconsistent.  吐出的豆子 = plural and in the second part, 那粒豆 "that one bean" singular.  Makes more sense to change it to plural] From then on, the village elders decided on a rule: No re-feeding the spat out beans back to the machine.  [You have changed the meaning of the original text]

对用机器的人，这事就这么了了；而造机器的家族得知此事，很是不安。Those using the machine, the matter was closed but when its inventors heard of this, they were unsettled.

他们无法接受这部机器有缺陷——虽然这并不影响使用。The inventors cannot bear to have a machine with such a flaw even though the flaw does not affect the use of the machine.

他们希望造出完美的机器，They hope to perfect the machine 能够象原来一样喂什么料吐什么豆，with the original concept - spat out beans no matter what was fed into it 同时不会因为喂了自己吐出的豆而陷入困境。including the beans that was spat out.

许多最聪明的人呕心沥血地探求解决方案，Many a smart person worked to their bones to find a solution 但一直没有满意的结果 but nary a satisfactory result/outcome.——这部机器，叫形式逻辑。制造这个机器的家族，人称逻辑学家。Thus the machine is the applied aspect of formal logic whereas the inventors of the machine are purists of an idealized world. [This is the hardest part to translate because translating it literally as "the machine is called formal logic and those inventors are logicians" is confusing to the English ears.  You need to translate the line's meaning or intent.  This is due to cultural preferences in how things are expressed.]

Story of a Bean Spitting Machine

There was a village where a clever family built a great machine that can spit a different bean when feed with a different stuff. By this machine, the village became more prosperous with each passing day.

One day, a child fed it with a bean spat out by the machine itself; it had been working incessantly but spat noting out. Finally, the machine had to be taken apart to get the bean out. Since then, the elders set up a rule: Never feed with the beans spat by the machine itself.

For the people who utilize the machine, the matter was cleared; but the family who made the machine was very upset. They couldn’t accept that the machine with a flaw though it in fact doesn’t effect the use, and wanted a perfect solution that, don’t get into trouble when feed back a bean spat by itself meanwhile keep the original function that feed with different stuff to spit different bean.

Many smartest folks then had been trying to find a solution painstakingly, till vomited blood, but nary a satisfactory outcome; yet – the machine was called as formal logic. The family built the machine, known as logician.

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Note

This revision adopted a lot of good advices or expressions from Jeff’s correction above.

The essay in Chinese at here: http://www.douban.com/note/484511991/; it’s taken a metaphor of the classical logic issue, namely the antinomy of the liar and the consistency problem of a logic system. It was written for some discussion on the forum in which there were some arguments on the issue. What I attempted to imply was that the machine (logic) has been always useful to us though it has some “flaw.” And the logicians, of course, need to seek the perfect solution.