A house contractor is one who either builds a home entirely or is doing a decent size remodel.
It does not cover handy men, specialty contractors who can only do one thing (tile laying etc) = The scope is for General contractors.
Population of the USA = 300 Mn.
Avg members / house hold = assumption = 4
So number of house holds = 75 Mn.
To account for a lot of other variants of house holds with < 4 members = adding another 20% = rounding to 90 Mn.
Assuming that these houses were built in the last 30 years, New houses built / year = 3 Mn.
I am making an assumption that for every home that is built in a year there are another 4 homes that are remodelled / some work is done requiring a contractor.
So total capacity of work done in a year by General contractors is = 3 Mn new homes/ year + 12 Mn remodeling homes/year
A contractor can do 1 or 2 new homes in a year, plus 2 remodelling in a year.
For simplicity I will assume we will need 1.5 Mn contractors to do the 3 Mn new homes and 6 Mn rmodeling , and another 3 Mn contractors to do the rest of the 6 Mn remodeling.
In total we will need 1.5 Mn + 3 Mn = 4.5 Mn Contractors.