That is not true. Not fully true, and the true part is blown out of proportion by various populists (especially right-wing, who would like to replace what we have with USA model or worse).
Most basic health care is organized by the government and paid through taxes and social insurance (which is obligatory). Unfortunately it is not financed enough and it shows, more in some areas and less in others. GP access is quite good, especially in larger cities, unless someone didn't care to choose his 'first contact clinic' right. Those clinics are mostly private, but working on government contract. One can usually get a GP appointment within a week, often same day. Urgent GP appointments are available 24/7 through special 'holiday and night health care points'.
Things became worse when popular specialist help is needed. One needs a referral from his GP and may need to wait months for appointment. There is the point were people who can afford that, would often go private. That and dentists / orthodontist.
Big problems are in children psychiatry, mostly due to lack of funding.
Medicines are much cheaper that in USA. When prescribed by a doctor they are usually partially or even, in some specific cases, fully paid by government. That is not make it affordable for everyone that needs it, but it is not very bad.
When something very bad happens – serious accident, cancer, etc. then the public health care gives the most. Public hospitals will do what they can (with limited funding and overworked personnel) for free. People are not sent away because they are poor and won't have huge debt to pay just because they got sick.
There are private insurances, or rather subscriptions services. They used to give better access to basic health care that the public services, but recently they don't offer much more. And you must pay for the public service anyway. They usually totally fail in more serious case (chronic illness, cancer, serious accident) – one would get to and be treated by a public hospital too.
In short:
Pros:
health care is basically free for everybody by principle
GP access is good, and serious cases are handled quite well
medicines are available and prices are not horrendous
Cons:
not all the free health care is practically available, sometimes available appointments are months or years in the future
dentists, orthodontists – not really available via public health care and private options are expensive
doctors, nurses are other personnel are underpaid and overworked
there is a lot of bad PR around health car here – this doesn't help improving things