I know this is an old question but I find this topic interesting and thought I'd contribute.
Why not an electric plane? The YouTube channel Real Engineering answers that question here:
It's over 10 minutes long but they give a good summary of the problem in the first 2 minutes. To cut that down even shorter, it is the energy density of the batteries we have today.
Why not fuel cells instead of batteries? Fuel cells will get hot and need to be cooled. The air needs to come in at a rate sufficient to supply the oxygen required. The exhaust will likewise need a free flow into the air. So, you have a very hot fuel cell, very hot exhaust, a big fan to cool the fuel cell, pump in air, and presumably propel the craft. To assist in the air in and out while stationary a turbine can be placed on the exhaust to drive an inlet fan. Question, what do we need the fuel cell for? Right, to power the big fan at the front. That brings more mass to the plane with a motor and then problems with the fuels that I'll get to later on.
The ability to "cut out the middleman" so to speak was discovered with piston engine development during World War II. All sides of that conflict figured out that as piston engines grew the contribution of the engine exhaust to the thrust grew. With a turbocharger required to get the most performance, especially at greater altitudes, the piston engine in the middle wasn't helping much. By just dumping fuel into a combustion chamber, putting a turbine on the exhaust, and use that to drive a fan to feed the flames of the burning fuel, the exhaust alone could propel an airplane to high speeds and quite efficiently. This efficiency over a propeller improved with greater speeds and altitudes. Later on bypass fans and propellers driven by the turbine were added when speeds remained subsonic.
What's the goal of using an electric airplane? Why bother when jet engines work so well? Oh, right, to lower CO2 emissions. We can use a biofuel for that. But there is a problem with biofuel, as it is currently done this is an energy negative process. We can improve this process but that comes with other problems. Again, Real Engineering explains:
If corn ethanol is out, as is soybean oil, then what about hydrogen fuel? Real Engineering has a video on this too:
While it is focused on cars much of the problems addressed comes in producing and storing the fuel. One problem pointed out is that producing hydrogen is an energy negative process. This is not a deal killer as there is a benefit in being able to turn energy from the wind, sun, nuclear fission, or heat within the planet into a liquid fuel. This means biofuels are not killed by being energy negative either, but by the need for land, fresh water, and labor that would be better spent on crops for food and clothing fiber.
What it comes down to is that the jet engine is just fine, it is the fuel that needs to change. The most likely candidate is synthesized hydrocarbons. These are drop in replacements for existing petroleum fuels and can be produced in a way that is a net carbon neutral. Burning the fuel does release CO2 into the air but we know how to get that carbon back and build more hydrocarbon chains with carbon neutral energy.
In response to @MSalters comment about fuel cell heat I'll make some more observations on the practicality of a fuel cell airplane.
I'm not claiming that removing the heat from a fuel cell is a problem, only that it must be done. As MSalters points out we learned a lot about how to build efficient and high flying aircraft in the last 100 years. The Meredith Effect tells us how to get useful thrust from heat sources on aircraft. We know from using piston engines at high altitudes that a turbo charger is an excellent way to get the same performance at high altitudes as at sea level.
Combine the Meredith Effect with a turbocharger and the fuel cell is just a combustion chamber for a jet engine. If someone where to go through several iterations of optimizing thrust from the Meredith Effect and the fuel cell powering a fan or propeller then could we see the fuel cell optimized out? My guess is that is exactly what would happen.
Does someone want to argue that fuel cells can do better than a jet engine? A jet engine can get about 40% efficiency. We can get the same from a fuel cell... at sea level. What happens in thinner air? The fuel cell will need something to pump in the air, and something to remove the heat. Again a turbocharger is good for this. That turbocharger will take some of that energy from the fuel cell. If we run the fuel cell hot then the turbocharger gets more efficient. Will higher temperatures lower the fuel cell efficiency? Perhaps, but it could be made up by a more efficient turbocharger that feeds the fuel cell air.
What if we go further in sacrificing fuel cell efficiency to get more from the Meredith Effect, and better turbo charger performance? Do this a few more times and the fuel cell isn't in the picture any more.
One reason people like a fuel cell is because it can run on "green hydrogen", hydrogen from carbon free energy. Well, a jet engine can run on hydrogen too. If the fuel is "blue hydrogen", hydrogen from fossil fuels, then it's no more "green" than a jet engine burning kerosene. We know how to run an internal combustion engine from hydrogen so unless we can get significant gains in efficiency from the fuel cell, which is doubtful, then the fuel cell carries no benefit.
It may be possible to build a fully electric airplane but it's not likely to be any better in costs or CO2 emissions than a jet running on "green" synthesized fuels of some kind.