If I set my OLS regression up as follows:

yHousePrice = x1Lot Size + x2HomeSize + x3DummyVariableCity +
x3DummyVariableHouseType + Errors

It is clear that my regressors are endogenous and therefore I will run
into a multicollinearity problem. No?  What is the standard approach
to correct for that?

If I need to eschew a linear model, please explain why.

Regression can be problematic WHEN there is "Perfect" Collinearity.
When there is multicollinearity, your regression may be BLUE. One
problem that you may be referring to is that multicollinearity, which
is high correlation between coefficients of variables, suggests the
statistical significance of each coefficient may be false.

In this regression, as far as I can notice, the correlation between
variables is small. There may be some correlation between
DummyVariableCity and DummyVariableHouseType. High collinearity can be
observed when you consider variable from similar categories - say,
HouseSize and NumberOfRooms (... well, not a good example, but I hope
you got the gist of answer...)

If there is multicollinearity, one solution to correct is to eliminate
a variable from regression. If, with high chance, one variable can
predicts the other variable, other variable can be omitted to avoid
some trouble. Another solution may work for somesituation is to change
the broadness of variable.

For housing price, linear model should be fine unless your plotting
data suggests logarithmic increments or exponential increments.