What he said.Be REALLY careful of allowing internet access to a storage device on your home LAN. I don't really recommend it.
First things first, if you want a shared storage system that does not rely on a running computer, you'll need to get a NAS. (Network Attached Storage) Basically this is an IP addressable disk device. The better ones allow RAID configurations.
Once you have the NAS set up, you can access it by IP address. BUT your home LAN should have a class C network, which is not routable, thus not accessible from the Internet. To get to it, you'll need to set up Network Address Translation in you home router. You need to assign a static IP and map it to your private IP.
The PROBLEM with this is that once you do, the device is open to the world, not a good thing. So be prepared to set up a hardware firewall to protect it.
My advice is leave the NAS private, and use something like GoToMyPC to access it through a local device.