Can I live off post-docs?
It will be very difficult to do so. Several European countries have rules about how long someone can be employed in "government" positions without a permanent contract. For instance, in Germany, if you do not have a unbefristete (permanent) contract after a given time, you are required to leave university employment. Other countries almost certainly have similar rules, as mentioned in the comments.

Now, there are positions above the postdoc level where one can remain indefinitely. For instance, in Germany, there are positions like the Akademischer Rat or Oberingenieur or außerplanmäßige Professor, who act in a secondary role supporting the head of an institute. Such positions involve substantially more administrative work than a postdoc normally encounters, but are also not subject to the time limitations. In other countries, such as the US, you can have "soft-money" positions that go by titles such as "research assistant professor" that are not formal tenure-track positions, and are dependent on the supervising faculty and the RAP raising sufficient funds to "pay for itself." I know multiple people who have had careers in such positions.