No, it is by no means completely free, it is paid for, as I said, through the national insurance contribution in income taxation. But, as has been said elsewhere, you must take into consideration the fact that such a thing overrules the need for everyone to pay for health insurance separately. From what I understand, the contribution we make is less than is required to maintain an insurance plan.
Also, on the government being too financially strained to cope, the NHS in Britain was set up only a few years after the Second World War. The government then was not wealthy by any stretch of the imagination.