70-300 f2.8 lenses don't seem to exist. What are the reasons behind that?
Manufacturers produce what they think they can sell. It is possible to produce a F2.8 70-300 lens, but such a lens will have a very large front element (approximately 107mm or 4.2"). Lens with such large front elements are quite costly, and customers quite rightly expect almost fanatical attention to optical quality. Correcting the compromises of such a long zoom range would add yet more to the cost. That size front element also means the lens is quite heavy (i.e several pounds, not ounces). All these combine to mean it is something of a niche product.
That said, Sigma makes a 120-300mm F2.8 zoom. It is not quite 70-300, but is close, and has gained popularity with sports shooters. Why have the big DSLR manufacturers not made one? I think because they do not see a large enough incremental market - it is certainly possible.
There are no "technical limitations". (End of answer)
The answer why such a Lens is not available (AFAIK) is: size, weight, cost, popularity.
Here are some Prime Lenses using Canon throughout for an Apples to Apples comparison.
Canon EF 400mm f/2.8L IS II USM Lens - U$10K - 6.42" x 13.5" (16.31 x 34.29 cm) - 8.49 lb (3.85 kg)
Canon EF 300mm f/2.8L IS II USM Lens - U$6.1K - 5.0" x 9.8" (12.70 x 24.89 cm) - 5.17 lb (2.35 kg)
Canon EF 200mm f/2L IS USM Lens - U$5.7K - 5.0" x 8.2" (12.70 x 20.83 cm) - 5.6 lb (2.54 kg)
Canon EF 70-200mm f/2.8L IS II USM Lens - U$2K - 3.5" x 7.8" (8.89 x 19.81 cm) - 3.28 lb (1.49 kg)
See how large the Front Element is and how long it is. The 70-200 is likely less expensive not only due to extra effort to decrease the diameter (come up with a better design) but also due to the greater sales potential (more Focal Lengths for your money, but not the quality of a Prime).
Sigma 70-200mm f/2.8 EX DG APO OS HSM Lens with 2x Teleconverter Kit for Canon EF - U$1.3K - 3.4" x 7.8" (86.36 x 198.12 mm) - 3.15 lb (1.43 kg)
The Teleconverter doubles the effective focal length while only losing 2 stops of light. So you get 70-200mm f/2.8 and 140-400mm f/5.6 - for the size, weight and cost you're better off but don't expect better quality for so little.