Why do we not reinvent the journal system? I am very surprised that I could not find any similar question here.

It seems to me that researchers from all universities are willing to write papers about their research for free and hand them into a journal to gain reputation. On the other hand, some other researcher do voluntary check the paper in order to examine if the paper is acceptable for the journal. Therefore, the main work is done by researches.   

The remaining work for the journal is to offer a platform where researchers can communicate and making sure that the reviewer is selected anonymous and to bundle many articles to a journal. 

Now everyone has to pay a huge amounts for the papers, they are not accessible for free to the general public, even though that most researchers are financed by taxes and only the journal is making profit.

Now I wonder, should it not be possible to create a network page for researchers which contains methods to imitate the review process of a paper? So that all papers can be downloaded at the website for free. I guess most researchers would be very happy if everyone could read their work. Also the money that is spend on journals by universities could be spend to this huge network page instead in order to keep it running.
Please meet the Open Access movement. 
In the last twenty years, many scholars and librarians did try to address the problem you pose. 
It is, in fact, a huge issue, and things like this don't change overnight. 

The OA movement focused on two main strategies:


Publishing in peer reviewed open journals (gold open access): the idea is to create a brand new journal (or to change the model of an old one) which will provide articles free for the readers, without the current subscription model in which libraries (meaning, taxpayers) pay. The crucial factor of Gold OA is the presence of peer review: organize real journals costs a lot of money, and at the moment the major business model is APG (author processing charges), meaning that the author (often, the faculty behind it) pays for being published and cover the journal costs.
We're still in transition, and there are a lot of drawbacks: there are predatory publishers who try to scam authors, and big publishers offer the "open access option" charging huge fees (this is also called double dipping, because a hybrid (both Open and Closed) journal will receive the money from subscriptions and the money from the authors. It is important to remark, also, that big publishers make a lot of money with subscriptions and they are actively challenging the open access model. 
PLoS, for example, is one of the new Gold OA publishers. 
Self-archiving in repositories (green open access): it is the model of arXiv, Repec, and thousands of other repositories. They can be "institutional" or "disciplinary", and they accept mostly pre-prints, but also post-print articles. 


Of course, there are some experiments in the field:


PeerJ has a very interesting business and pricing model.
Epijournals (from my understanding) are journals created selecting papers from arXiv and other repositories. 


References


Open Access, by Peter Suber, MIT Press, 2012.
Wikipedia "Open Access" article. Accessed 4 Jan 2015.


Full disclosure:
I've worked as a digital librarian in managing OA journals from University of Bologna. I'm biased towards OA and open knowledge in general. Please keep it mind that my answers reflect these bias.