Your best bet would be an airplane/drone that is built from carbon composite and graphlite rod (pull-truded carbon rod with incredible strength - widely used in sailplane spars) powered by a couple of aviation diesels like the Austro Engine E4 .
The Specific Fuel Consumption of the E4 is .336
lbs/hp/hr.
Except for similar competing diesels, nothing in any traditional aircraft engine configuration comes close except the old tubro-compound mechanical-monstrosity radials like the R3350 that could get into the high 3s (normal gas piston engines are around .45 lb/hp/hr and turboprops well above that, around .5-.6 lbs/hp/hr).
The E4 is 168 Hp so two would give you 336 HP for take off, and the engine burns 5 US Gal per side at 60% cruise, or 30 lbs per side, so to go 2000 nm, if you can get the thing to cruise at 150kt, you need 130 US Gal capacity plus a VFR reserve, say 140 gal total, or 840 lbs of fuel.
So design an airframe that can haul, say, a thousand pounds of payload with those two engines and 840 lbs of kerosene, and is clean enough to go 150kt, and there you have it.
aviation.stackexchange.com/users/30504/John K
It's perfectly possible to build piston-engined planes with ranges of 2000 miles or more, for instance many WWII planes like the B-17 and B-29, pre-jet airliners like the Lockheed Constellation, or the Rutan Voyager.
The reason there are few, if any, in service today is that there is no real market.
Jets have taken the long-range commercial market, and pilots of smaller planes generally don't want to go that far without stopping for bathroom breaks.
If you are looking to design an unpiloted plane, the problem becomes even easier.
You should be able to use modern, non-certified engines derived from the automotive market, can fly at efficient speeds, and don't have to worry about breaks.
aviation.stackexchange.com/users/6968/jamesqf
This is perhaps not going to be a very good answer to the specific question, but the mission you describe would constitute some out-of-the-box thinking.
As John K already suggested, a very light construction, perhaps in the spirit of Rutan Voyager suggested by Jamesqf would be a good starting point.
Large wingspan to get efficient wings would give the chance to use less powerfull and/or exotic engines.
John K's example of Austros is a good one.
In my mind what you should design is a motorglider on steroids.
Relatively large cargo would otherwise require a lot of thrust, but you could compensate with wingspan and 2 or even three lifting surfaces.
Leaving the pilot out would save at least 400 lbs of weight (pilot, seat, controls, gauges and such), plus you would not need any pressurising, so you could fly ballistic profile flights: high power ascents (but not that steep
as exess thrust is not great with full cargo), and the gliding the rest of the way.
You could even feather the props and shut down the engine -> no cruise consumption (piston engines typically are not very efficient when operating in partial load = cruise).
I don't think speed is of essence here, as even a cruise (or in the case I'm proposing, glide) speed of say, 100 kts would be, knowing the infrastructure there , at least ten times faster.
I see one real problem though: landing at destination, automatic or remotely controlled?
If rc, how?
Satellite would be the only aplicable solution, but might prove troublesome with videostream bandwidth, lag etc.
If automatic, oh man the effort of programming relating to all that needs to be taken into account.
So: is dropping the cargo by parachute out of the question?
aviation.stackexchange.com/users/42636/Jpe61
The main distance limitation for a plane is weight: every extra pound of fuel means one less pound of payload.
Planes can be equipped with "ferry tanks" that increase endurance for long open water legs, but then they're bumping up against (or even over) maximum gross weight with just the pilot (no passengers or cargo).
Take the pilot out and you can add more fuel to go even further.
Unless your goal is moving the plane itself, though, having no weight left for payload doesn't really allow anything useful to be accomplished.
Note that aircraft range is often specified for something like 75% power at cruise.
Since drag is proportional to speed squared, flying slower would use a lot less fuel and thus extend your range, which may be fine for a drone that lacks the human need to stop every few hours.
aviation.stackexchange.com/users/35166/StephenS
2000 miles with a piston engine has been done more recently - check Wikipedia's entry for the AAI Aerosonde, which crossed the Atlantic (2031 miles) burning only 1.5 US gallons of fuel.
It includes a reference to the later "Spirit of Butts' Farm" which did that as well with a total gross weight of 11 pounds / 5 kilograms.
Autopilots are far more capable and small than they were then and often include autoland capability.
What you want to do can be done.
aviation.stackexchange.com/users/31011/JimHorn
Ok so Im exploring ways to bring cargo and medical supplies to various rural cities in say Africa without relying on ground infrastructure. Just short and cheap runways spread around the continent. It seems any type of VTOL plane wouldn't have the range. Electric is also out. It also seems like piston planes don't have the range, while turbojets are millions of dollars. What about used turbojets? Whats stopping piston engines from flying longer?
Can you fly 2000 miles on an engine that costs less than $500k? If so, what type of engine?
aviation.stackexchange.com
<jet-engine><aircraft-performance><engine><airplane>
aviation.stackexchange.com/questions/74810
aviation.stackexchange.com/users/46981/Aiden Kashi
It's perfectly possible to build piston-engined planes with ranges of 2000 miles or more, for instance many WWII planes like the B-17 and B-29, pre-jet airliners like the Lockheed Constellation, or the Rutan Voyager.  It is also possible to use a couple of aviation diesels like the  Austro Engine E4.  If you are looking to design an unpiloted plane, the problem becomes even easier.  Moreover, you should be able to use modern, non-certified engines derived from the automotive market, can fly at efficient speeds, and don't have to worry about breaks. A good solution would be to use carbon composites and graphite, paired with aviation diesels, to design a new airplane or drone. You could also consider designing a light piston-engined construction with large wingspan. The Rutan Voyager or a powerful motorglide might be a good source of inspiration to this end. Finally, autopilots could be a good idea, as they are small and efficient, and offer additional advantages like autoland or the possibility to use non-certified engines.
A good solution would be to use carbon composites and graphite, paired with aviation diesels, to design a new airplane or drone. Consider designing a light piston-engined construction with large wingspan. In this context, an autopilot might be easier to build, as you could use non-certified engines. The Rutan Voyager or a powerful motorglide might be a good source of inspiration to this end. Autopilots could be a good idea, as they are small and efficient, and offer additional advantages like autoland or the possibility to use non-certified engines.