This question might look strange, But I hope it make sense.
What should I do to host my site at one than one locations (shared or dedicated)? I would like to avoid down time, so I thought I will keep the copy of the site in 1+ servers, and when one hosting is down the site will be automatically fetched from the next server?
Is it possible to sign up with 2+ shared hosting, setup multiple DNSs and make the site downtime nil? Is yes, what is the mechanism to do this?
There is only a few domain registrar which allow maximum 4 nameservers. Generally it to be 2 means nameserver1 and nameserver2. Hosting one website in multiple server is only possible if domain registrar allows multiple sever name setting in your domain registration account. As for as I know more than 2 name server is not allowed in most of the companies.
Is it possible to sign up with 2+ shared hosting, setup multiple DNSs and make the site downtime nil?
It is possible to host a copy of your site on two or more servers. Third party DNS is something to consider. Use the search terms "round robin dns" or "dns load balancing" in any search engine to get how to information.
As far as making the "downtime nil" - that is probably a pipe dream but look up the given terms and see what you think.