eBay and look for used Dell servers. I'd go for a Dell R2x0 series for a starter box. Maybe a used Dell Workstation if you want a tower.
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You might be able to find an HP DL360 for around that price
Go on eBay, punch in the price you're looking to spend and search for an old server. Keep in mind some manufacturers use proprietary connectors.
Look for servers with lots of ECC ram, clean photos of the internals.
They probably won't have a drive that's pretty common.
To meet that 16 core requirement, you'll probably be looking at older dual socket systems.
Edit: a quick search I found this. https://www.ebay.com/itm/225978893065?
Not a perfect match but the price is pretty good.
I wouldn't worry about a specific number of cores, as that doesn't translate at all to CPU performance. Instead look at single thread performance.
So something with a Core i5-7500 on ebay for example, should be able to get one in your price range after buying more RAM.