The Museum of RetroTech proudly presents a sack-barrow powered by a small petrol engine.
Left: Motorised Sack-barrow: 1960
But can you buy a powered sack-barrow nowadays? You certainly can, from the Handle-It company. Interestingly all of their powered barrows are specified as 'stair-climbing' which seems to imply you're a bit of a wimp if you want a powered barrow just for moving stuff on the flat. The site talks about a 'motor' so presumably these barrows are electric-powered with a battery, but that nowhere seems to be stated. They are expensive: £2,294.00 + VAT for a basic model.
There is another company making them here. Once again the motive power appears to be battery-electric, though it is not stated.
And there are more to be seen at BritishBins who are not shy of saying that their barrows are electrically driven and rechargeable.
No modern petrol-driven sack barrows have been found so far, but you can certainly buy petrol-powered mini-dumpers.