Is there any way to keep someone from accessing information on your computer using their computer if you share the same network?

Make sure Sharing is turned off.

Yes.  Turn on your computer when it is the only one being on.  Turn off your computer before any others on the network are turned on.  The network hardware has no ability to store information if all the computers attached are off.

Perhaps some exotic hacking program might be able to reside in a networked device other than a computer.  A printer's RAM or settings, a router's BIOS, or a modem's memory might be a transfer method, but I have not heard of that (yet).

The answer is "firewall"

I would be the first to admit, I am not an expert in this area. Many virus protection programs are equipped with a firewall to block unwanted access to your computer. Also, to access your computer, the hacker often needs to trick you into downloading some sort of malware to your computer. So if you have a firewall, and you are diligent in watching what you donwload to your computer, and you have a robust password to access your computer you are generally safe. Additionally, the perpetrator has to have some expertise in hacking a computer, this is not a common talent.