Hi, I'm filming a zero-budget short film and I need some footage of
either a Dog-Race or Horse-racing. The longer the better, I've found a
few sites but all require a huge amount of cash, which being a student
film-maker I dont have access too. Any help will be greatly
appriecated.

If you don't mind old footage, you might want to try the Internet
Archive.  They have TONS of old newsreel type footage.

Here is one example, a 33-second horse racing clip titled "Willie
Shoemaker at Golden Gate Fields."
http://www.archive.org/movies/details-db.php?collection=prelinger&collectionid=jockey

Here is the main page:
http://www.archive.org/movies/movies.php

The ephemeral collection is especially good.  The clips can be
downloaded in a variety of formats and sizes, so you should be able to
incorporate them into your work fairly easilly.

Hope this helps!

Try the National Archives in DC. They have the Universal Newsreel from
1930 to 1960 and contains many horse race clips. You can either visit
or ask them to research for you, it will cost you the price of dubbing
the tape format you require but the cost will be nothing compared to
the regular commercial libraries.

Try www.archives.gov