There are several elements required in order for your website to be accessible:
1.) your webserver must be up and running.
2.) there must be a working network route providing connectivity to and from your website.
3.) your website's name (fully qualified domain name) must resolve to the correct IP address.
Backup solutions for each of those differ:
1.) You can harden your webserver (mirrored disks, redundant power supplies on separate power feeds) but as long as it is a single server there is always going to be some single point of failure. Getting a second server to publish the same content seems like the obvious solution but is really only simple if all website content is static. When working with a hosting company instead of running your own server consider choosing one that is able to replicate your server (ideally in two different locations, see #2).
2.) Multi-homing your webserver with Internet connectivity from two providers significantly reduces the probability of loosing network connectivity. However if both providers use fibers or copper pairs in the same cable bundle a single bad dig will still cut you off. Your odds are better if you go for multiple webservers (see #1) and further place them at different locations (still go with different Internet service providers). If you are using a hosting provider (instead of running your own servers) you can ask them for redundant network feeds (some have this option and even give you a choice of providers) or better yet setup a redundant webserver with a different hosting provider (this is the most complex option since it leaves it your own responsibility to synchronize content between the two servers).
3.) Name resolution is done with DNS. There is nothing wrong with using the DNS servers from the domain name registry you used to register your domain name. If they are following Internet Standards and BCPs they will have multiple servers in geographically separate locations. However it would still be a good idea to have another DNS server for your domain (either one you are running yourself or a 3rd party providing secondary DNS) just in case your registry does something like GoDaddy did (preventing access to all their DNS servers at once).--
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