Equipment wise, I have three approaches for very quiet sound sources: Contact mics (picking up mechanical vibrations rather than air), a large condenser mic (inherently lower self-noise than small condensers), and Sennheiser MKH mics (lowest noise mics I can afford).
Technique-wise, more than half of it's listening. When I hear tiny, delicate sounds, I grab them, regardless of what makes them, because really subtle sounds can be used for many different things. Fluorescent lights that produce a rolling series of clicks as they're turned on are my favorites.
Selcuk provides really important advice, though: Sound design is a mix of capturing, editing, and recontextualization. Making it sound right doesn't mean making it sound accurate. This also takes the pressure off of trying to record the impossible.
Georgi is also painfully correct: Rooms, and even what's outside those rooms, starts to matter a whole lot once the sound source gets super quiet. Can't tell you how much I cringe when I can hear exterior noise in the backgrounds of what I record indoors. :-(
All this said, there are scientific devices and techniques that record infrasound and ultrasound, but the recordists I know don't have access to such gear.