Simply using an external drive is not a good idea. For the great majority of purposes, hardware failure is a vulnerability and prying eyes are not.
Your files aren’t safe from loss unless you have multiple copies in multiple locations. Also, services like Dropbox have versioning – so if you accidentally overwrite or delete a file, you can retrieve a previous version.
If you are concerned having your files on someone else’s computer, you can add encryption to Dropbox using BoxCryptor
Or you can create your own Dropbox-like service.
Make a private dropbox with an old computer in minutes
Because music and photos take up lots of space, I wouldn’t waste your Dropbox allotment on them.
For music I use Google Music Manager. It stores 20,000 songs for free. Also, you can sign in from any web browser and listen to your music.
For pictures, I use Google Picasa. Flickr is a good option, too.
For other files I use Dropbox. I have used Box.com, too. They both work in the background automatically.