I graduated with my PhD in math two years ago, and I am no longer affiliated with a university. However I am still actively researching. Often it's possible to get papers on arxiv or directly from the author's website, but sometimes it isn't (particularly for older papers). Public libraries, at least around here, don't have subscriptions to mathscinet. Do I really have to pay if I want these papers?
Another option is to simply visit a university campus. The bigger the campus, the better. I would just go to one of their libraries, maybe even their central library. Connect to their wifi and then you should have access to journals and papers.
I also know of some university libraries where you can become a member even if you are not a student. Sometimes the memberships are free but sometimes you have to pay like an annual fee.
Further, you might be able to do something with your university where you graduated from. Plenty of universities have alumni services. My university has an alumni club with a ton of benefits but the only thing I use is access to the library which is way bigger/useful than a public library could ever be. In the last two of the cases, you should be able to use VPN to just log-in remotely from home and get the papers whenever you want instead of physically visiting a campus every time you need to read papers.
Mathematicians will often send you their preprints (not the work in progress), if not already on their pages, if you ask kindly. I suggest to title the email with "Paper request: [Paper title], [Journal name], [Year]" for an efficient query.
Back in 1996, I was working my Msc on a subtopic of number theory (diophantine approximations and continued fractions of polynomial roots in finite fields). I was away from my university, and working without a university student card. I was spending a lot of time at Institut Henri Poincaré, photocopying articles from mathematical journals for my research. I badly needed a conference paper: A. J. van der Poorten, Continued fractions of formal power series, Advances in number theory, Proc. of the 3rd conf. of the CNTA, 1991. No university around had these proceedings, and Australia was too far.
"Alf" was a big name in the field (to me at least). And email and WWW were not very widespread in France. Yet I found his email, and spent about 1 hour to write the most polite email I could, in the afternoon. The next morning, the postscript was in my inbox, with a nice answer by A. J.: "thank you for your interest in my work, good luck with your study."
This gesture both:
which helps me a lot now. I am still thankful to him now, and often think about him, even if I never met him. So far, mathematicians have always answered my requests, sometimes even with a scan and standard mail (with nice stamps) when the paper does not exist in electronic form.
Now, a lot of papers are available online. I insist one should browse researcher's webpage pages (instead of relying only on web search): sometimes, old papers are present in formats (dvi, ps.gz) that are not always indexed.
Finally, this could lead to discussions and, who knows, collaborations.
If everything else fails, different options exist. One with twitter hashtags has made the news, I only mention it for the sake of information, and do not endorse the legal aspects.
Are you doing research alone? Or do you still have contacts with your advisor or a collaborator at a university? If you work with an academic, speak with them; maybe they can find a way to get you an account to access the university network via a VPN.
This could involve giving you a meaningless title such as "external research associate" or "external collaborator". As long as there is a faculty member vouching for you, and it's free for the university...
There is/was an unofficial way to use Twitter to crowdsource papers. Here's how I was told it works:
Now I know some people were using this to illegally get papers they hadn't paid for. These people would delete their first tweet after getting the emailed pdf. There's also apparently a rule where you're not supposed to thank the sender either. I don't quite understand that one.
I don't know if this is still a thing, but you shouldn't use this to get papers you aren't legally allowed to have. That would be immoral and deprive the hard working journals of their extremely reasonable and low priced access fees.