Poor man's anycast: Setup a hidden master and push zone updates to multiple slave DNS servers at different providers, which are all registered as your domain's DNS servers (at your registrar).

Actual anycast: Piggy back on a network provider that already announces an anycast network block and "lease" a single IP address. Have lots of time and money? Obtain BGP access, colocate servers, purchase network equipment, etc. and do it all yourself.

A good compromise is to use a DNS provider that offers anycast already, like DNS Made Easy, AWS Route 53, or a number of others at reasonable cost.