Not hosting it on a dedicated server. There is no way around the fact that if you have something on a machine and that machine is not available, that something is not available. This is basic common-sense logic.
Now, depending on what your website does, there are easy ways. Static content? Run the access through a CDN and have the CDN refresh regularly.
Dynamic content (LITTLE dynamic, like a news magazine)? Same - except the CDN can be instructed to serve from the server if available and use the last served version when not.
Full dynamic (like a shop)? Realize that what you want is irrelevant - we are back to "one item not available means service not available. The first 2 solutions worked on the same principle, but you could outsource the multiple redundant machines to the CDN - that does not work for i.e. an event booking service.
Seamless browsing is not discussed in many scenarios because physics and cause and effect are still things - if you run a single point of failure, it is YOU deciding that and the universe cannot magically work around that.
So, get more machines. Or start moving your stuff to a cloud service. You may find that suddenly cloud is not that expensive, depending on what you consider a server. Not being snippy here - I just find that even lower mid-range servers these days are terribly powerful for a simple website, especially if you offload static content to a CDN. With very few exceptions, the only reason to run a physical server these days is to run virtual machines. Note that I consider "mid-range" a dual socket system - not your homemade thig running a desktop CPU. And yes, that is a problem - for reasons if security, I want a separate machine controlling my backups and I have trouble finding something low power enough and anything higher is price inefficient. But this is how things are. So, you may find that cloud services are way more efficient than 3 servers - and yes, you need 3 as things sometimes break.