Actually, you'd be better off not building an oxygen storage unit, but to build a reversible glucose-oxygen fuel cell. This would have the advantage that when it was doing its thing, not only would oxygen be supplied and $\text{CO}_2$ removed, but glucose would also be supplied, making the recipient almost completely self-sufficient as long as its power supply held out.
If we have $$\text{C}_6\text{H}_{12}\text{O}_6 + 6\text{O}_2 \to 6\text{CO}_2 + 6\text{H}_2\text{O},\quad\Delta G = −2880 \text{ kJ per mol of } \text{C}_6\text{H}_{12}\text{O}_6$$ then by reversing this and applying energy, we can turn carbon dioxide and water back into glucose and oxygen, using power stored in the converter implant or supplied externally via whatever means (induction?)
If we go by current trends in battery technology, battery capacity doubles each 22 years. From 2015 to 2050 is 35 years, so we'd expect 235/22 = 3 times the capacity in our best batteries. Current Li-ion batteries are 460kJ/kg or 827kJ/L, so we'd expect 1380kJ/kg or 2481kJ/L.
Given that the volume specified is 'half a lung', the total volume of the lungs is about 6.16l in an average human: (Total Lung Capacity + Physiologic Dead Volume), so half of one of two lungs would be about 1.54L.
Considering that our battery/fuel cell might have 0.04L of ancillary equipment and be 2/3 energy storage by volume, we'd have 1L of battery, capable of storing 2481kJ.
If we take a highly active 18-30 year old, 90 kg male, from charts on this page, we can see that the daily energy requirement is up to 18.8 MJ/day.
This means that 2.418MJ divided by 18.8MJ/day = 0.129 day, or 3 hours. By reducing energy consumption with total inactivity, this could conceivably be stretched out to six hours.
Recovery time would basically be the recharge time for the battery - it could be charged by using the glucose-oxygen fuel cell, or by external induction. There would be some waste heat, but the human body is very good at dealing with waste heat. It might recharge in as little as a quarter hour on external power, up to an hour if relying on glucose and oxygen.
It is not inconceivable that a reversible glucose/oxygen fuel cell could be invented by 2050, though it would be pretty bleeding-edge technology.