In git, is there a simple way of introducing an unrelated branch to a repository?

The currently selected answer is correct, I would just add that coincidentally...

This is actually exactly how github.com lets users create Github Pages for their repos, thru an orphaned branch called gh-pages. The pretty steps are given and explained here:

https://help.github.com/articles/creating-project-pages-manually

Basically the git commands to set this up are as follows:

1. git checkout --orphan gh-pages (create a parentless branch called gh-pages on your repo)
2. git rm -rf . (removes any files from branch working tree)
3. rm '.gitignore' (even the gitignore)
4. Now add website content (add index.html, etc) and commit and push.
5. Profit.

Note you can also designate a /docs folder on you repo to be the source of the "Project Site" that Github uses to build the website.

Hope this helps!

Although the solution with git symbolic-ref and removing index works, it might be conceptually cleaner to create new repository

$ cd /path/to/unrelated
$ git init
[edit and add files]
$ git add .
$ git commit -m "Initial commit of unrelated"
[master (root-commit) 2a665f6] Initial commit of unrelated
 1 files changed, 1 insertions(+), 0 deletions(-)
 create mode 100644 foo

then fetch from it

$ cd /path/to/repo
$ git fetch /path/to/unrelated master:unrelated-branch
warning: no common commits
remote: Counting objects: 3, done.
Unpacking objects: 100% (3/3), done.
remote: Total 3 (delta 0), reused 0 (delta 0)
From /path/to/unrelated
 * [new branch]      master     -> unrelated-branch

Now you can delete /path/to/unrelated

From Git Community Book:

git symbolic-ref HEAD refs/heads/newbranch 
rm .git/index 
git clean -fdx 
<do work> 
git add your files 
git commit -m 'Initial commit'

There is a new feature (since V1.7.2) which makes this task a little more high-level than what's in any of the other answers.

git checkout now supports the --orphan option. From the man page:

git checkout [-q] [-f] [-m] --orphan <new_branch> [<start_point>]

Create a new orphan branch, named <new_branch>, started from <start_point> and switch to it. The first commit made on this new branch will have no parents and it will be the root of a new history totally disconnected from all the other branches and commits.

This doesn't do exactly what the asker wanted, because it populates the index and the working tree from <start_point> (since this is, after all, a checkout command). The only other action necessary is to remove any unwanted items from the working tree and index. Unfortunately, git reset --hard doesn't work, but git rm -rf . can be used instead (I believe this is equivalent to rm .git/index; git clean -fdx given in other answers).

---

In summary:

git checkout --orphan newbranch
git rm -rf .
<do work>
git add your files
git commit -m 'Initial commit'

I left <start_point> unspecified because it defaults to HEAD, and we don't really care anyway. This sequence does essentially the same thing as the command sequence in Artem's answer, just without resorting to scary plumbing commands.