The reason why it is not possible to install OS X Mountain Lion 10.8 on an Asus EeePC 1005HA is because it uses the Intel Atom N270 processor that is 32bit only while OS X Mountain Lion is a full 64 bit operating system only.
In fact it’s possible to find an ‘hackintosh’ version of the OS X Mountain Lion 10.8.1 kernel for Atom support on the OSx86.net project:
http://www.osx86.net/view/3160-mach_kernel_atom_1081.zip.html
but that would be compatible only with the 64bit versions of the Atom processors:
http://en.wikipedia.org/wiki/Intel_atom#64_Bit_Support
Instead, you can follow these instructions to Install Mac OS X Snow Leopard 10.6.7 on the Asus Eee PC 1005HA.