If you have access to Windows fonts, the recently announced Bahnschrift looks to be Microsoft's interpretation of the DIN outlines. In some time it will be available on many computers running Windows and could well be a good, free and close representation of what paid DIN fonts offer.

Our new first OpenType Variable Font Bahnschrift is now included in builds. This new font industry standard enables us to pack an entire typeface family into a single file with infinite variability. No longer are you constrained to simple weights like Light, Regular, and Bold. Now you can have an infinite range of font styles, with smooth interpolation from Light to Bold and beyond. Better still: because a single, efficient variable font can replace several static fonts, variable fonts save a lot of space.

There's an open source DIN-inspired font called Gidole:

"Oswald" font is a pretty good alternative. It's a google web font, so it's the best possibility for websites ;-)

https://www.google.com/fonts/specimen/Oswald

TRY roboto .It looks almost like DIN and it comes in so many styles . Its feels really premium. http://www.fontsquirrel.com/fonts/roboto

Titillium Web is also a free alternative:
http://www.google.com/fonts/specimen/Titillium+Web

DIN Next from Linotype would be the commercial version with webfonts available:
http://www.myfonts.com/fonts/linotype/din-next/

I like Gesta as a stand-in for DIN. It's not actually all that similar (Gesta is more stylized) but the x-height and overall mood seem to match very well.

http://r-typography.com/gesta.html
(webfont on Typekit)

John's suggestion of Abel seems good, too. I haven't compared the fonts very closely, but it's got a DIN feel to it at non-display text sizes with a character width sitting somewhere between Engschrift and Mittelschrift.

http://www.madtype.com/tag/abel/
(webfont on Google Web Fonts)

I have used Babel Sans as an alternative to DIN: http://www.dafont.com/babel-sans.font

Although it is bit thin.

Cabin is a pretty good free alternative to DIN. Well, it may be fairly different in design but bear with me - sometimes it can feel as if it is similar:

DIN:

Cabin:

Dosis is a pretty good free alternative to FF DIN Round if you were interested in a rounded version.

Also, League Gothic could be a decent alternative to DIN Engschrift (ie DIN "compressed") though it has slightly less of a "German" feel to it.

Here's a bolder version inspired by DIN:

ROPA SANS - https://fonts.google.com/specimen/Ropa+Sans

Try Gesta from Typekit. I's very similar. https://typekit.com/fonts/gesta

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.