In fact, it is being done, in South Korea. You can check out the now weird normality of a country which has successfully suppressed COVID-19 without shutting down at this link. It is well worth a read.
Interesting to compare their figures to ours: at 9,876 cases in a population of 51 million SK has 0.019% infection rate. At today's date we have 708 cases in a population of 4.5 million for a 0.016% infection rate. Let's hope that our initial lockdown gives time for us to catch up to their levels of testing, tracking, and tracing - and then we can get back to something more like normality.
This is the shape of the SK curve.
You can check out at this excellent dashboard by Johns Hopkin University and Medicine. Also see the post below on modelling the impacts of the virus.