The only way I found to do it is by installing a 3rd party file explorer software. Isn't there a default app that does that? At least for media files?
Certain apps like the gallery let your share content via Bluetooth (or any other number of methods). The gallery handles video and images. Highlight the item and hit menu to get the Share menu.
Bluetooth File Transfer provides OBEX FTP and OPP for android. It allows you to send a file via bluetooth to an android device or any other devices that implement these bluetooth profiles.
The BT stack on Android phones does not support the file transfer profile. You may be able to add this by rooting your phone and loading a custom BT stack.