Well, first things first, does your old computer have a USB port on it, Silencer?
If it does, there are a couple of things you can try.
One, you can try and purchase a mini USB drive (small enough to fit a pocket) if you can find one, around where I am, they usually hold up to 4 GB of data. You just need to plug one in to the USB slot (if you have one) to be able to use one of these.
Two, if you have bigger files you want to keep (e.g. if they are at least a 100 MB or more each), then an external USB hard drive is better, but generally more expensive. They can carry much more space than the pocket sized versions (I've seen some on Dell that are quite expensive, but can carry at least 1 TB of data).
If not, then, about the only thing I can suggest for backing up important files of yours would be floppy disks (for the smallest files, mainly text files), or CD-RW's (but only if your old computer has the power to burn files onto a CD).
Either way, hope that helps.