The ability to estimate the internal weather conditions of shipping containers globally without using sensors will enable new monitoring and risk assessment solutions. We tackle this problem by developing both linear and nonlinear regression models to predict the internal temperatures and relative humidity of containers from meteorological data. Our training data consists of sensor measurements from 85 shipments across 3 continents over 4 months - the largest data ever collected in an academic publication. We extract features from 761 gigabytes of weather data and incorporate physics (psychrometry) models to engineer new features that boost the performance of our models. Our best models make temperature and relative humidity predictions with mean absolute errors of $1.8 ^{\circ}\mathrm{C}$ and 5.0% - measurements well within the uncertainty range of the sensors - and thereby demonstrate accurate sensorless monitoring.