No, the are not even nearly airtight, and leak air quickly on their own.
Air pressure is maintained with bleed air or a powered pump. FAA has a thorough, well-illustrated guide to the systems involved.
As for why they're not built completely airtight, aircraft are mostly riveted and glued, not welded, so their structure is inherently not solid. You'd need to use sealants throughout the structure, install and maintain seals on every opening, and compartmentalize a few systems that need outside air by design.
That's extra weight and maintenance, which doesn't have to be there, as cargo doesn't need pressure and humans need fresh airflow. The minimum rate is ~1.6 L/s/pax for survival and ~8 L/s/pax for comfort. More flow means more O2, giving more consciousness time if decompression happens (breathing low ppO2 greatly accelerates the onset of hypoxia), less CO2 (affecting cognitive ability and comfort), and less humidity, so that less odor is perceived and less water accumulates in the insulation.
With outflow valves closed, airliners normally lose about 400 feet (~5%) of their pressure difference per minute when new, this varies per airframe. The permitted limit is typically 750 FPM. High-flying aircraft like A380, A350, 787 are built with considerably lower leak rates, a safety margin that allows them to get certified above FL400 by providing more time to get to 10k ft in emergency.
As aircraft age, their leak rate can double and more, usually being close to the limit for an average airliner. A common offender is window frames rattling themselves a bit of clearance. When tested and found to exceed the limits, maintenance is required to bring the leak rate down. So even the current level of airtightness takes effort to maintain.