As a keen avocational palaeontologist I often run into the problem of finding papers, but there are ways around this:

1) If you have a reference or abstract, contact the author directly. Workers are often very happy to send a copy of their papers.

2) Open-access journals: These are becoming increasingly more popular as an alternative to JStor etc. Authors are realising they need to get their research to a wider audience and are increasingly publishing through these outlets which are mostly online. Check forums and blogs in your chosen subject area, ask for copies of papers or links if necessary.

3) Libraries often keep copies of journals or can get them for you.