Liquid Oxygen
Oxygen is stored as a very cold liquid in a container very similar to a thermos. When released, the liquid converts to a gas and you breathe it in just like the compressed gas.
This storage method takes up less space than the compressed gas cylinder, and you can transfer the liquid to a small, portable reservoir at home. Liquid oxygen is more expensive than the compressed gas, and the reservoir vents when not in use (see safety issues).
An oxygen-conserving device may be built into the reservoir to conserve the oxygen.