How to play MP3s on iPhone 5s without iTunes?

Contrary to the other answer, the current version of dropbox supports playback of MP3 files. Starring a file, i.e. marking it as a favorite, will download it to your device and make it available offline, so you will be able to listen to it without an Internet connection. Mega also supports playing MP3 audio on IOS and offers a bit more free space than dropbox.

What you won't be able to do is have your MP3 files appear in the Music app.

As for playlists and playing more than a single file at a time you'll need a third-party cloud streaming app, such as CloudBeats (paid, supports other cloud storage services besides dropbox) which worked for me but I didn't like it enough to want to buy it. There are also other apps in the appstore if you search for "streaming player" In any case, imo, this is not the way to go. I choose to stream all my music collection via Google Music - works equally well on IOS, Android or a desktop web browser.

This lifehacker article has a review of audio streaming solutions.

You cannot do this from the Dropbox app. Neither can you transfer music from your Dropbox account into the Music app directly (without getting them to a computer, adding to iTunes and syncing the music to the iPhone).

You can play songs from your Dropbox account using other apps. Search for "dropbox music player ios" on the web to find some of these apps. Tunebox, BoxyTunes and DropTunes are a few (paid) apps that can do this.

You could try Tiny Player (author here; app is free). It lets you load MP3 into your phone over WiFi from a browser or a small companion Mac app.