This is a very useful question. It has 5 different helpful answers that say quite different but complementary things (surprising, eh?). This answer combines those answers into a more useful form as well as adding two more solutions.
There is no Oracle Express Edition for 64 bit Windows. See this official [but unanswered] forum thread.
Therefore, these are the classes of solutions:
- Pay. The paid versions of Oracle (Standard/Enterprise) support 64-bit Windows.
- Hack. Many people have successfully installed the 32 bit Oracle XE software on 64 bit Windows. This blog post seems to be the one most often cited as helpful. This is unsupported, of course, and session trace is known to fail. But for many folks this is a good solution.
- VM. If your goal is simply to run Oracle on a 64 bit Windows machine, then running Oracle in a Virtual Machine may be a good solution. VirtualBox is a natural choice because it's free and Oracle provides pre-configured VMs with Oracle DB installed. VMWare or other virtualization systems work equally well.
- Develop only. Many users want Oracle XE just to learn Oracle or to test an application with Oracle. If that's your requirement, then Oracle Enterprise Edition (including support for 64-bit Windows) is free "only for the purpose of developing, testing, prototyping and demonstrating your application".