Why do you require so much bandwidth? Is that what your actual usage is or do you just assume it will be that much? Bandwidth gets quite expensive, depending on the datacenter in use, so be sure that you actually need that much.
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You can get that much bandwidth included if you buy a dedicated server. I do not know how many VPS providers will offer that much Bandwidth with a VPS. I am sure there are some out there though. I would think a Dedicated Server would be a better fit...
You need a Gbit VPS so you can "hope" for 100mbs speeds. And with $50/month and all of the other requirements its hard.. Have you tried your luck by searching in the advertisement area ?
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4TB translates to about 12-13 Mbps _average_ monthly speed. Providers need to purchase on peak, not on average. It will be hard to get that much real bandwidth with your current budget. Maybe you can get close if it is only He.net or Cogent bandwidth (those are the cheapest options).