Ideally I'd have a custom webserver where people can upload and share, but I don't have the skillz or the time to do that really.
I think Woogley answered this wish before it was even uttered. Google Base may well fit this bill. There's a Java API, it's not going anywhere anytime soon, unparalleled uptime and bandwidth.
Downsides are: Users would have to have their own Google logins if you want to avoid astonishing opportunities for mischief for malicious users, and AFAICS there's no way to upload files to Google Base through the API, so you'd have to split the data into the contents of several attribute tags. I don't recall the exact limit on the length of each attribute, but I'm pretty sure it wasn't cripplingly miserly.
If I were in your shoes I reckon I'd go with Moogle's suggestion - store the data invisibly (maybe java.util.prefs?) as a matter of course, but give the user the option to export it to a nice simple file somewhere obvious like the desktop. Presto! No filechooser needed.
user.home on linux works as expected, giving you the /home/user directory. Linux users will expect anything saved there withour their explicit
consent to be in one of them hidden directories like /home/user/.game_data