iPhone is broken, how can I still access my contacts?
If you synced them through iCloud, then the easiest way is to go to  iCloud. From there you can go to Contacts, select them all using the gear menu on the bottom left and export the vCards. You also can open iTunes and see if it has already been set to sync contacts to an app on the computer or if iTunes has a backup that you could restore onto any iPod, iPhone or iPad to access them / cloud sync them.

Otherwise if you were backing up the device on your computer you can recover the data with iPhone Backup Extractor.

Or, if the phone still boots up (e.g. you just broke the screen or something like that), you can also try using some utilities like PhoneView (I'm pretty sure there are others, but I've used only PhoneView so I don't know if/how they works).