Yes, it could. You could compress the air at your house using an air compressor, fill a compressed-air tank in the car, and the car could run off of it. You could use an engine very similar to a steam engine (using pressurized air instead of pressurized steam) to convert the compressed air to rotational energy.
The main problem you would have is the size of the "gas tank" you would need to get any kind of reasonable range. You would need a LOT of compressed air to take a car very far.
An alternative would be to use a liquefied gas like liquid nitrogen. You can think of liquid nitrogen as air that has been compressed so much it has liquefied. Liquid nitrogen is much denser than compressed air, so you could store enough in a car to give it reasonable range. You could then fill up at the liquid-nitrogen station.
You are right that a compressed air car would generate no pollution while you are driving. However, the power plant that produces the electricity that compresses the air (or liquifies the nitrogen, for that matter) is probably generating pollution from the coal or oil it is burning. Power plants are relatively clean and very efficient compared to car engines, so this is not a bad trade-off. Still, it's important to compare apples to apples.
Another thing that would be interesting to figure out is the efficiency of a liquid-nitrogen engine. If you burn gasoline in a car engine, only about 20 percent of it powers the wheels. The rest creates wasted heat. It would be interesting to see how efficient a power plant is, plus how efficient a nitrogen liquefier is, plus how efficient a liquid-nitrogen engine is -- that would tell you whether you were creating a technology that is better or worse than the gasoline engine in terms of efficiency.
Here are some interesting links: