The situation has changed since this question was first asked in 2012.

Alte Din

There is now an OFL-licensed, completely free/libre version of DIN called Alte DIN.

This is legal because DIN 1451 is a product of the German government, and so is in the public domain — only the individual interpretations of it by various font foundries are protected and copyrighted. Thankfully Peter Weigel traced it for us!

D-DIN

A corporation called ‘Datto’ commissioned Monotype creative type director Charles Nix^[1] to create a DIN-like font which was released in 2018 under the SIL OFL 1.1.

As of 2021, it is no longer available on their website, but you can find it on a reputable site like FontSquirrel or GitHub. “Altinn” (Github) is a fork of this project on Github. “Dinish” (Website)(Github) is a fork of this project with continued development, which has also been submitted to Google Fonts.

Original answer (2012):

I don’t know how close you need to come to the DIN typeface, but I found a couple possibles on Google Web Fonts. Using the letters aGgQqlJ to narrow down the letter shape matches, I found:

Wire One – https://fonts.google.com/specimen/Wire+One

The lowercase letter shapes are quite similar. The overall font is a bit more more condensed than DIN and some of the capital letter shapes are different.

Abel – https://fonts.google.com/specimen/Abel

Also, not an exact match but the feel is quite similar.