Anycast in IPv4 just means that the IP address used is present on multiple machines and so can be reached in multiple places, not having to get back to a common end-point.
If you're not planning on getting other networks to use your anycast instance, then you don't need to announce the route and so don't need to use BGP or have an AS. You just need to make sure that your internal network has routing which provides a route to the anycast IP which is local.
Eg, if you have a border router and hang the anycast IP on a box which is on a switch connected to that, then you might have the anycast IP configured on loopback on the hosting box and have the router use a static route which points the anycast IP at the normal IP of the hosting box.
Within your organisation, you control the routing and you don't need to register what you're doing. This is commonly used for implementing things like blackhole routes, for instance. In practice, the common expectation is that you don't hijack IP address space not assigned to you; you don't create debugging problems for others; you don't interfere with the ability of others to be reached on the Internet. As long as what you're doing isn't breaking anyone but yourself, though, it's your network to do with as you please.
For an example of address-space anyone can use, you might look at http://www.as112.net/ which has three IP addresses that anyone can implement on their own network.