I want to develop something similar to jsfiddle in where the user can input some data and then \"save\" it and get a unique random looking url that loads that data.
It's an odd set of constraints. I routinely use MD5 checksums to generate unique URLs from data. If the user doesn't already have the data, they can't guess the URLs.
I do understand about not wanting to use a database—if you've never used one before, the learning curve can be a little steep.
I don't understand the constraint about "storing things sequentially on the server." If you need to know the order in which the hashes are created, I'd simply put that information in a separate file. You might have to do file locking or some other kind of hack to make sure you can append a hash to that file incrementally.
If you want short URLs, you can either take a prefix of an MD5 checksum or you can take a CRC-32 and base64 encode it. Both will give you unique URLs with reasonably good probability.