Well, A records are, by definition, exclusive. MX records realize you might need multiple mail servers and are the only DNS record that is weighted.
The only way you could set up a website to do failover would be to use a load balancer.
I'm sorry but that's not correct. You can set multiple resource records into one resource record set for example one entry like www with roundrobin. So it is possible to achieve some sort of load balancing via DNS based load balancing.
Extending this with healthchecks if the corresponding server is available and if not change the config automatically is a solution that is used quite often for financial reasons since you don't need additional load balancing equipment.
Of course this solution is primitive and not capable for all scenarios but it works.
bind is the first address to achieve this.http://ftp.isc.org/isc/bind9/cur/9.8/doc/arm/Bv9ARM.pdflhttp://www.google.de/search?source=ig&hl=de&rlz=&=&q=DNS+based+load+balancing+bind&aq=f&aqi=&aql=&oq=