We present the first set of detailed molecular-line maps of the region associated with the giant Herbig-Haro flow HH 300 from the young star IRAS 04239+2436. Our results indicate that the red lobe of the HH 300 flow is depositing a fair amount of momentum [3.2(sin i)^-1 M km s^-1] and kinetic energy [2.6(sin i)^-2 × 10⁴³ ergs] over a notable volume (~11%) of its host dark cloud. This makes HH 300 a key player in the evolution and fate of its parent cloud. The redshifted molecular outflow lobe of HH 300 is 1.1 pc long and 0.3 pc wide and has a very clumpy structure. The density, velocity, and momentum distributions in the outflow indicate that the observed clumps arise from the prompt entrainment of ambient gas. Bow shock-shaped structures are associated with the clumps, and we hypothesize that the shocks are produced by different mass ejection episodes. Lines drawn from IRAS 04239+2436 to each of these clumps have different orientations on the plane of the sky, and we conclude that HH 300 is a precessing and episodic outflow. The observations include a map of the red lobe in the CO (2-1) line, with a beam size of 27'', and more extended maps of the outflow region in the ¹²CO (1-0) and ¹³CO (1-0) lines, with 45'' and 47'' beam sizes, respectively. Due to "contamination" by emission from another molecular cloud along the same line of sight, we are not able to study the blueshifted lobe of HH 300. The combined ¹²CO (1-0) and ¹³CO (1-0) line observations enable us to estimate the outflow mass accounting for the velocity-dependent opacity of the ¹²CO (1-0) line. This method is much more precise than using ¹²CO data alone. We obtain a steep power-law mass spectrum for HH 300, which we believe is best explained by the evolution of the outflow mass kinematics. In addition, our ¹³CO (1-0) observations show that the HH 300 flow has been able to redistribute (in space and velocity) considerable amounts of its surrounding medium-density (~10³ cm^-3) gas.