Why is the 737 so low?
The 737 was originally designed to be a smaller aircraft serving more regional routes. The original JT8D low-bypass turbofans easily fit under the wings, and allowing the plane to sit lower to the ground made it easier for the plane to operate at smaller airports without support equipment. Passengers could use the built-in air stairs and the cargo bay was low enough that ground workers could climb in to load and unload baggage.

The low height also allowed shorter and lighter landing gear. The 727 had tail-mounted engines which allowed it to sit lower to the ground, so keeping the 737 low helped to keep some commonality there as well. Over time the airplane has grown longer, been fitted with larger diameter engines, and become capable of much longer flights.

The A320 was designed from the start for the larger family of CFM56 high-bypass engines. The main competition was the 737 Classic version, which was powered by similar engines and had grown substantially from the original 737 versions. Both planes ended up in a market where the low height of the 737 didn't have as many advantages.