I am getting lots of downvotes on my question because its poorly researched (as claimed by downvoters). I want to know how to research.
I read/watched book/movie decades ago. And, I don't have access to reliable websites to do research. What to do?
Should I leave this community?
If you've permanently moved to this new country that doesn't have access to Google or Wikipedia, that will severely limit the resources that you have available. However, this is the internet we are talking about, you should be able to find other websites/search engines that will aid you in researching an answer.
That being said, every answer has a certain layer of subjectivity, and if you feel that you aren't providing answers that have good enough references, the answer seems simple: Don't answer them then.
If I see for example, a Harry Potter question to which I think there is a canonical answer from the book, but don't have the reference in front of me to be accurate, I won't answer until I do. From your other questions 1,2 you're worried about down votes, and from this one particularly about providing accurate research/sources for your answers.
Really this has to be up to you, but it seems like you feel that you can't provide decent sources for answers anymore, nor do you feel that you can properly research a question. If that is the case, then it is likely that your questions/answers will continue to get some down votes. (Keep in mind that as far as rep goes it takes only one up vote to counteract five down votes on an answer.)
According to your comment you say that you have are currently in a different country, could you elaborate as far as which country you are now in? There are plenty of resources available aside from Google or Wikipedia depending on what country you are in. Search Engine Colossus has a list of the best available search engines for a large collection of countries, hopefully that will provide you with sufficient resources to continue contributing quality answer/questions.
Have you tried:
?
There is more to SF&F on the web than Wikipedia and Google.
There are many proxy sites one can use, as well as VPN sites, such as StrongVPN.com, that allow one to create private connections and tunnels to other sites or to freely access the rest of the Internet. Some proxy services are free, some have a subscription fee.
If the country you are currently in is blocking websites, and you don't want to use proxies to get around this I see two options that may remain legal:
Contact a friend in another country who is able to provide you with information. Or...
Head to a local library to find the local source of information.
These could both be an issue if the country has locked down these options as well, but I would certainly move before letting a country's laws force me to leave an online community such as stack exchange, though I know that is a very personal decision.
