In most cases, the constant in the OLS model doesn't make any sense in reality. Because if we'd like to interpret the meaning of the constant, we set all regressors to zero. For instance, we want to estimate a regression of a house's price on their sizes of lot in feet including constant; and apparently it's impossible to find such a house with no size of lot.
Although the constant loses the interpretation, it plays a critical role in obtaining an unbiased estimator. In order to get the "true" estimator, we want to minimise the sum of square error and hence this process usually returns a model with intercept. The model without constant tends to get larger error. Here's a graph of a simple linear regression model from the wikipedia. But if we draw a straight line through the origin, it could be seen that the direction of the regression line would completely change and how larger the error would be.