Is there a maximum amount of data that a Drupal database can safely store? Is there a maximum amount of data that a Drupal, let's say MySQL, database can safely store? If it gets too big will problems start to occur?
The short answer is no. There is not reasonable way you can store "too much data".

If you read the Limitations entry on wikipedia about MySQL, you'll see that "maximum amount of data" doesn't come into the picture.

More specifically, MySQLs documentation says


  The effective maximum table size for MySQL databases is usually determined by operating system constraints on file sizes, not by MySQL internal limits.


Depending on what you do with the data, however, you can get performance problems if you store a lot of data, but that's not a Drupal specific problem.