How do I export an Android Studio project?

As mentioned by other answers, as of now android studio does not provide this out of the box. However, there are ways to do this easily.

As mentioned by @Elad Lavi, you should consider cloud hosting of your source code. Checkout github, bitbucket, gitlab, etc. All these provide private repositories, some free, some not.

If all you want is to just zip the sources, you can achieve this using git's git archive. Here are the steps:

git init       # on the root of the project folder
git add .      # note: android studio already created .gitignore
git commit -m 'ready to zip sources'
git archive HEAD --format=zip > /tmp/archive.zip

Note: If you intend to send this by email, you have to remove gradlew.bat from zip file.

Both these solutions are possible thanks to VCS like git.

In the Android Studio go to File then Close Project. Then take the folder (in the workspace folder) of the project and copy it to a flash memory or whatever. Then when you get comfortable at home, copy this folder in the workspace folder you've already created, open the Android Studio and go to File then Open and import this project into your workspace.

The problem you have with this is that you're searching for the wrong term here, because in Android, exporting a project means compiling it to .apk file (not exporting the project). Import/Export is used for the .apk management, what you need is Open/Close project, the other thing is just copy/paste.