Is it possible to replicate the experience gained in a grad school on my own without attending one? Is it possible to become a researcher on my own? How do I replicate the grad school experience through self study or being an autodidact?

I came across these two sites: How to Become a Pure Mathematician (or Statistician) and How to become a
GOOD Theoretical Physicist. 

I came across such a thing on CS Theory Stack Exchange too.

Is it true that if I follow actively what has been given in these sites I’d be a physicist or mathematician or an independent researcher? I don’t know about publishing papers. At the best, I can access arXiv and Google Scholar on my internet connection. I don’t think I’d be actively ‘contributing’ to research or advancing the frontiers of knowledge. At the best, I guess I can try answer questions on SE sites.
The short answer is: you can't replicate it.

The longer answer is: yes, you might be able to talk intelligently to someone who has a background in the field, but as previous poster mentioned, the reason why one goes to grad school is not just to learn the material, but to have access to resources. These resources are physical, but they are also mental. Part of what grad school gives you is access to people who have extensive knowledge in the field, can help direct you in your research, as well as provide crucial feedback. Without this one-on-one and group interaction, it is IMO impossible to become a researcher on par with a professional academic.