tl;dr: use Dropbox …

Dropbox is a transparent to access cloud storage provider that can be used to easily sync files across devices, with platform support for Linux, Mac, Windows, iPhone, Android and also a web interface. It is free to use for the basic account which gives you up to 2 gigabytes of storage. Computer versions can be found here: https://www.dropbox.com/desktop

The beauty of using Dropbox is it will transparently (in the app on e.g. your phone) play the mp3 and remember where you have got to next time you access it.

Here are the steps I used to listen to mp3 audiobooks on my phone:

1. Obtain mp3 of audiobook
2. Place into Dropbox folder, this is probably easiest done on a computer
3. Install Dropbox app on your phone if you have not already. Note that you can set specific files and/or folders to be available offline within your Dropbox app on your phone; this works by downloading these files and storing them on the device, which is best done when you are connected to wifi
4. Play 🎵