Save a process' memory for later use?

This is usually called a persistent continuation. Some languages like SmallTalk and SBCL have first class support for persistent continuations. Most languages don't.

Depending on your requirements and OS you could try forcing a core dump

I have never tried actually loading a core dumped program back up other than in gdb. It seems like any files you have open or any other state that is not in your programs memory would be lost as sharptooth pointed out.

Another approach would be simply serializing the state you need to disk in your program. It sucks but it is probably the most reliable way unless you are content with suspending execution of the program. That could be done with your operating system's thread library. Or as one poster pointed out with your shell.

Technically it is possible, but it would require saving all the system-allocated resources state too - like file descriptors for example and then restoring them. So it's a challenging task.

The easiest way to achieve what you want is to use a virtual machine like VMWare. When you pause it you actually save the whole machine state together with all programs running.