Power transmission/AC or DC/Cable cross section/Converters
At which Voltage should I transmit this power?



As high as possible to minimize the cable cross section, since that will give you the lowest copper cross section. At some point the insulation will start to dominate unless the wires are physically separated. Let's suppose you use 350VDC. 


What cable should I use? What can be the smallest cross section?



Something rated to be safe at 350VDC. You have not divulged any environmental or regulatory limitations, so let's say it's just regular wire rated for 240VAC mains. 

The current will be 1300W/(0.8 * 350) assuming 80% efficiency in your converter, or a bit under 5A (that's ignoring wire losses). The wire size calculation is another question and it depends a lot on the conditions. Note that the current might be higher if the loss in the wire turns out to be high, since you'll have to pump more in at one end to get it out the other. Let's suppose you use AWG20 wire with 200°C PTFE insulation. Resistance should be maybe 8 ohms (you will need to find the self-heating and do a calculation if you want it to be accurate). That means the voltage loss at 5A will be 40V. So the current now is more like 5.6A, not too far off. 


Should it be AC or DC transmission?



Doesn't much matter for only 100m, but let's say DC because you get more power for a given wire size and peak voltage rating. 


What will be estimated size of the final converter (I need 24V DC at the end).



An Artesyn model UFE1300-5 should do it. 272.8mm x 140.0mm x 40.6mm, would be an estimate of the size. Weight is 2.5kg for the converter alone. Add maybe 1.2kg for the wire
and you're up to 3.7 kg. 2.5~3kg is typical for 1300~1500W DC-DC converters. 

The above is just a bunch of rough guesses without a bunch of important information to make a reliable and safe design, but from the information you gave, it does not appear to be too unreasonable a goal for an R&amp;D project, but perhaps not quite off-the-shelf.