yes its possible but not advised
you could set an IPv6 subnet to use privately -
http://www.simpledns.com/private-ipv6.aspx
however your router will need to do IPv6 to IPv4 tunnelling - not usually available in consumer grade routers and its still NAT which is one of the reasons IPv6 was created - to eliminate NAT - its not actually designed for NAT and trying to do IPv6 NAT is not recommended
to be honest an IPv6 only network is only really useful for testing - theres absolutely no real benefits compared to the negatives at this stage - IPv4 will remain around for quite some time despite the IPv4 address pool being all but exhausted and IPv6 adoption is only a few percent at this time
if you want to play with IPv6 you might want to try playing with tunnelbroker.net - you could use your PC to set it up if your router is not compatible with it its free and would give you IPv6 on your LAN (dont disable IPv4 on your LAN it will break too much stuff) and ability to browse IPv6 sites along side the current IPv4 sites