Are there any public and big family trees available online as downloadable gedcom files?
Something like royal families, presidents or like that.
If you're looking for big family trees of royal families, presidents, etc. in GEDCOM format, my favorite site is Famous Family Trees by Paul Pruitt.
Paul has all sorts of neat trees, many which he created or contracted for, including:
The main purpose for his site is to house Paul's Catalog of Life Database, which includes 1,257,735 different living species that inhabit the world. I believe this is the largest meaningful GEDCOM ever created and it includes 2.1 million individuals. It is 650,134 KB in size.
The site also has interesting GEDCOMs for a few fictional family trees (Harry Potter, Lord of the Rings, the Simpsons), a couple dozen Corporate Family Trees (including General Electric, Walt Disney, Apple and others), Genetic Haplotypes, the Structure of the US Government, a family tree of Human and Computer Languages, and the Microsoft Operating System Version History.
Update Nov 1, 2022:
As Jeremy Jones noted in his comment, it appears that the website genealogyoflife.com is no longer active. That is where all the GEDCOMs I mentioned were stored.
However, it does appear that many of the GEDCOMs were saved by the Internet Archive. You can find a listing of the ones available here: https://web.archive.org/web//http://genealogyoflife.com/tng/gedcom/
Tamura Jones has a gedcom generator "GedFan" that can be used for testing purposes. A blog describing latest version is at 2015-03-18 GedFan 0.4.0.0 See the links section at the bottom of that page for other resources.
Although each individual is bare-bones, the number of individuals is determined by the number of generations specified.
I use a GEDcom from this source: Thomas Stobie's Genealogy Pages for testing high-volume imports -- no famous people, but it's public and big ("640,000+ people")...