Here's a quick take that I will attempt to evolve into a better, more-XKCD-style answer over time:
I believe it's possible, but planning is required.
Trying to do this with fully-loaded planes isn't a good idea; you are more likely to succeed with reduced fuel (perhaps no fuel in the towed plane) and cargo.
This gives you excess thrust available in the towing plane, and every pound less of lift saves some fraction of a pound in towing force (I'm having trouble finding 737-specific numbers right now but Lift / Drag ratios of 15 to 20 aren't uncommon for an airliner and would change over the members of the 737 family).
The towing plane will have to generate both the thrust it needs and the thrust the towed plane would need, plus some additional from the towing bridle not being perfectly straight and taut. For 737s specifically this gets a little iffy just based on the ratio between maximum takeoff weight (MTOW) and empty weight; I don't see any model of the 737 with that ratio as high as 2.0. Perhaps you'd need to reduce the weight of the towed aircraft even more than "empty" (i.e. by removing engines).
There is every reason to believe that the towed aircraft could be rigged in such a way to receive the force, because the engine pylons and wing boxes already have to resist that force and more. Presumably we will be towing at speeds slower than cruising speed to buy ourselves less overall drag and better Lift / Drag ratios. Rigging the towing aircraft may be more complicated; my first guess would be attaching cables to the wing box at the wing roots, but that's a guess.
It will help if the towed aircraft can remain under control to maintain a good following attitude, use additional lift controls like slats and flaps during appropriate phases of the flight, retract and extend the landing gear, etc. It is not clear to me at this time if a towed 737 would have to have a running engine to do so, or if an APU would suffice, or if newer models (or other aircraft like a 787) could run their controls entirely from an electrical connection from the towing aircraft (which could end up being one of the costliest things to engineer in this whole proposal).
I still don't see any reason to believe that particularly long runways should be necessary. Though a 737+737 combo will likely not be able to operate out of all the runways a single 737 would, part of the point of long runways is to be able to abort takeoffs on the ground and we'll assume that among the operating controls that the towed aircraft has, it has working brakes.