You have multiple solutions.
On a pure DNS level you can publish your records with a low TTL (say 5 minutes), and have your monitoring systems change the content of the zone by removing the dead record when detected. This does not provide immediate fail-over but can be often good enough.
It does not involve too complicated systems.
Also, some DNS servers allow some "programmed part", with a dynamic backend that can compute records based on some external parameters, like doing live checks and replying only with the live records.
Anycast is another solution indeed, and has then no relationship with the DNS anymore (although the DNS itself can be "anycasted" but then it is to resolve its possible failover needs, not the ones of your application).
Basically your multiple systems, on various places in the world, are advertised with the same IP address. So the DNS has only one record.
With the "magic" of BGP, each instance announcing a given IP address will collect all the nearby traffic, so you get load-balancing for free in fact. And you need some specific tooling so that, as soon as some local instance is dead (or in maintenance mode for example), you stop announcing its IP address there, so that all other networks in the world, again because of BGP, learn that to reach "something" behing that IP they need to go somewhere else, to another instance of yours announcing this IP.
This is far more complicated to setup as you need a proven BGP setup (and making errors in BGP can have even greater consequences than in DNS), and multiple instances located in different datacentres, and possibly multiple AS numbers, depending on how you want to do your anycast done. This clearly needs skilled professional in BGP routing where the first solution with only DNS (in the first case of just changing a static zonefile) is reachable by any enthousiastic amateur.
So the answer also slightly depend on the network locations of your load-balancers.
