You can use round robin DNS, which is the term for what you're describing, but that just means that when one host is down, 50% of the requests will timeout. DNS isn't the solution to this problem.
You can balance a site across any number of back-end servers. You need a loadbalancer/reverse proxy to handle this. The problem is that if you aren't hosting yourself or don't have a colo for your own equipment, you'd need to outsource this load balancing just like you're outsourcing your web servers and the load balancer becomes the additional point of failure.
If a situation where people colo or self-host and really care about uptime, there are usually a pair of clustered load balancers with inbound connections from multiple ISPs feeding into them, so that you've done your best to eliminate single points of failure. Combine this with anycast and load balance clusters and servers hosted around the world in multiple datacenters, and you have some pretty good availability.
Now, of course, all of that is way overkill for you. You should probably just switch to a host that doesn't suck, but the process above is important to understand. Even if that understanding is just "way more complicated than I can afford right now."