Elementals don’t need to breathe, eat, or sleep, thanks to their Elemental Nature (MM p. 123). This means that one could survive indefinitely in a Bag of Holding.
An Air Elemental using its Air Form ability could enter a creature’s space and stop there, effectively engulfing a PC in air. While the statblock does not clarify if this is breathable air or not, there is also no such thing as "breathable air" in 5e; things that you could breathe that are not air tend to be Poisons or specific gasses, not "air." The convention, then, would be to assume that the "air" in an Air Elemental was breathable. In addition, the statblock of the Air Elemental does NOT say that the creature sharing the space of the elemental is unable to breathe or begins to suffocate, implying that the creature inside the elemental is fully capable of breathing normally.
Could you, therefore, put a party member and a friendly elemental into a Bag of Holding (assuming that there is enough rations in the bag of holding to tend to the character’s biological needs) and expect them to survive indefinitely?