What is a dependable machine to dig a hole 18 inches wide and a loader at the same time?
Your budget is on the low side for anything but a used machine of dubious quality. I don't think on 5 acres you're going to do enough digging to justify owning a backhoe. I'd suggest spending $10,000 on a good used diesel tractor with a front loader and a 3 point hitch in back (you'll use that for operating mowers, rototillers, or numerous other kinds of equipment on the farm), and spend the other $2,000 paying another farmer in the neighborhood that has a backhoe to come do your digging for you as needed.
Five acres doesn't warrant a full grown Diesel tractor. Take a look at some of the compact tractors like Kubota or Yanmar. You can probably find a good used or reconditioned one with loader and backhoe in your price range.
Bobcat makes an excellent mini tractor with the attachments that you will need. It can have a loader, back hoe, and has a P.T.O. for other attachments as well.