The BuiltLean program is based on movement patterns, so it would fall into the “functional exercise” category. The only equipment you need to do the program are dumbbells, a pull up bar, and a bench.
While you don't have to have a gym membership, it can definitely make it easier because you'll have all of the equipment you need to do the workouts.
If you want to incorporate other weight lifting equipment into the program (like barbells or kettlebells), you'll just have to substitute the exercises. For example, instead of doing an overhead press with dumbbells, you could do a kettlebell overhead press instead. Or similarly, instead of doing dumbbell goblet squats, you could use a kettlebell or barbell. We just recommend that you have a strong background of experience with these more advanced pieces of equipment.