There are some great deals on servers on many sites (eBay being one of them). We've picked up some great bargains where businesses are removing "end of support" servers that are immaculate. Sure, they're not the the latest models and therefore may not be the most powerful (nor efficient) but there's plenty of life left in them.

There will be a number of refurb businesses that also offer used servers that have removed equipment similar to described above and even offer warranty on them.

I would have thought that most datacentres would entertain sourcing the server spec you need, but I could be wrong there.

As per the other replies, some datacentres may not wish to entertain a desktop type case, or if they do, you'd need to pay additional rack space on them to cover the additional U of rack used.