Yes, you can run ubuntu on a USB Flash Drive.
I did a search up at the top of the Newbie forum on "flash drive" and on "uSB stick"
Tons of threads on how to do that -- heres a linkhttp://www.pendrivelinux.com/
So yes you can do that, but the fact remains that you could have a bad Hard Drive and you need to get that fixed to get full use out of your Mini-Laptop.
Again, do a search in the newbie forum, read the threads for the HowTo stuff and gollow the bouncing ball.