Will a website recover from a Denial of Service attack on its own?
You should stop your screen-scraping software if you have not already.
Depending on what part of the system is down (Either the database, the server, the network or all), there is a chance it will recover by itself when the loads come back down.
If your application cannot sustain 250 simultaneous connections, you will want to investigate why. The culprit is usually database load (no indexes, un-optimized queries).
Linode could also have restrictions in place to limit how much bandwidth can be used within a certain period of time. You should probably contact them (or whoever is in charge).
I'm in a rather peculiar situation right now.  To make a long story short, I'm part of a (real life) volunteer organization of about 2000 members.  Our current website was built and maintained by a member who is no longer part of the organization (he quit).  Unfortunately, he was the only one that actually had access to the server, and hasn't been cooperative in handing over the reigns to someone else after he left.  As a result, myself and a small team of people have been working on creating a new website for ourselves from scratch.  The data on the original website would be awesome to have for the new one, so without direct access to the database we have been screen-scraping what we need.
Which brings me to my current conundrum.  The screen-scraping script I was using was being really slow, so I had the brilliant (not) idea of parallelizing it.  I assumed the bottleneck was caused by my slow internet, so I foolishly decided to run 250 threads at once.  After I tried that, the web server mysteriously went down and hasn't gone up since (it's been about 30 minutes now).
I'm not any kind of hacker or security expert, but I'm pretty sure I just accidentally caused a Denial of Service attack on the server.  Which brings me to my question - assuming the owner of the website does nothing to help us, will the server come back to life of its own accord? (it's being hosted by Django on Linode if that matters).  How do websites typically recover from DoS attacks?  Have I potentially misdiagnosed what's going on, and could there be an alternative explanation?  Or is the website lost forever?
Edit:  All 250 of the requests were simple http requests going to pages within the Django admin panel if that changes anything.