I have been using the Google Cloud Storage Manager
link on the Google APIs console in order to upload my files.
This works great for most files: 1KB, 10
In my experience, the accepted answer is not correct - maybe it was but something has changed.
I just uploaded a file of size 2.2GB to GCS using the web interface on Chrome 42 on Windows 8.1.
I would also point out that the question is about files > 2GB, and the answer mentions 2GB, but gets that from 2^32, which is 4GB, not 2. So maybe the limit really is 2^32 (4GB) - I haven't tried anything that big.
(It is still a good idea to use gsutil for large files.)
The web UI only supports uploads smaller than 2^32 bytes (4 GigaBytes). I believe this is a javascript limitation.
If you need to transfer many or large files consider using gsutil:
gsutil -m cp /path/to/*thousands-of-files* gs://my-bucket/*