The broader issue as I see it, is that for a layperson or a hobbyist in a scientific field it is near impossible to find papers that you did not hear about through the media or by being in proximity to the authors.
If you do happen upon them, it's usually by pure dumb luck and happenstance. More centralized and open locations for discourse are required.
How about a ycombinator for scientific research?
o.
The trouble is that services needs to be domain specific and it is likely a full time job to continually mine the journals for new content. User submissions are an excellent idea, but probably not sufficient on their own. For example, there is already HN-like site for biomed: http://hackermed.com/hot
There is opportunity here, especially in the areas like health and fitness which are more popular to the public.
1) Publications are increasingly easy to find via services such as google scholar, pubmed, and citeseer. Often times, these services will link to freely downloadable versions of the paper. However, should this not be the case, you still have two options.
2) Go to the author's website. It's very common for academics to put preprints or other free versions of their papers on their personal websites. You may need to check all the authors on the paper, but a good rule of thumb is to check the first listed author and the last listed author.
3) If the above isn't available and you really want to get access to the paper, you can always try emailing the authors and asking for a copy. As with personal copies on websites, giving out copies to those who ask is allowed by most journals.
4) As you point out, you can physically go to a library and use a public terminal. Most large cities have universities. Save up the hard to find references and make a weekly or monthly trip to the local university library.
In the far past, you used to have to go down into the "stacks" at a university and dig out a physical copy of a paper you wanted. These stacks were not always accessible to the public and did not always contain the journal you wanted. In comparison, academic material today is more accessible to the average member of the public than it has ever been.