These might be off the usual suggestions.

BTW, how big are the files you need hosted? A few dozen gigabytes at most?

If you want something new, you can opt for an Asus EEE desktop PC (or similar offerings from other PC makers or even make your own low cost PC based on ITX or entry-level all-in-one ATX motherboards) without screen. You can use a KMV (keyboard/mouse/video) switch to use your existing PC's peripherals to access the system when needed. These PCs consume very little power as compared to regular desktop units.

Or if you have an extra old laptop not being used anymore, you can have the OS reinstalled (or upgraded) and configured for file server purposes.

A NAS is good if the data to be hosted is to be accessed within the LAN only. But if it needs to be accessed via WAN as well, it might not be the solution you would like to use.