If your existing HDD is 90% empty, and you only need to grab like 10-20Gb of stuff from it you can get a flash drive, load ubuntu as you said and just burn the files to a CD/dvd. or alternatively you could pop that HDD into another computer and do the same thing of burning data to DVD (say go to a friend or something).
if that drive has majority of the files you need (over 100Gb), then since you are already short on storage, you should go ahead and buy another internal drive. Put it in, make it your OS drive and then pop in the old one as a secondary, grab files from it, reformat it and then use it for storage.
I think that's the best reasonable course of action