I would suggest to get an external hard disk and put your files on that one. I have my files on the computer, on an external hard disk and on a usb stick. Don't just save them as midifiles, since when you do some things could get changed when you'd open them with notation software. You have finale right? Save them as finale files, as xml and as midifiles, so your fonts, chord symbols etc won't get lost. In midi files that can happen.
There also options to save files online but personally I feel not good about doing so.
I still have old floppy disks which I can not use anymore since my old computer broke down. I hope your computer has the possibility to connect to another hard disk since that is what you need to do.