I use an NFC access card. Is there any way to clone my card using Galaxy Nexus / Nexus 7?
See the edits for recent updates to this although the answer is still a general 'no'
In short, no, not easily anyway.
You can get all of the information off it by using an app such as NFC TagInfo however currently stock Android phones don't support emulating NFC cards even though their hardware allows for it.
If you were willing to install a custom ROM on the device then there may be a working ROM that allows for the functionality on those devices, there is one for the Nexus S, however these ROMs usually only let the phone act as a tag so you can't put data on it directly from the phone, you'd have to write the data onto it from another source.
I also haven't seen one that allows for the setting of the UID (Unique Identifier) which is what most door readers and access control systems will work with so even if you could emulate the information on a tag you still couldn't open the door as the UID would be wrong.
EDIT: It is now possible to load applications onto your device and have them communicate with a reader via NFC however for this to work you must be running Cyanogenmod and this still doesn't solve your problem, the app still can't change the UID that the phone shows.
EDIT EDIT: With Android KitKat 4.4 the ability to have the phone act as an NFC card is built into the operating system so you no longer have to rely on Cyanogenmod. However it can't actually emulate a tag (such as a MiFare tag) which is what most access cards are and it can't set its own UID. n.b. I haven't had a chance to play with this much so I'm just going on what I've read.