The largest passive tag I've ever even heard rumours of could store a whopping 8192 bytes. (8KB.) And that was "wait, are you making this of gold?!"-grade expensive.
If you want something that's self-powered and has its own processor, NFC peer-to-peer essentially allows for any size you care to stand there and transfer. At a maximum transfer rate of 424Kbit/s, however, you're going to be standing there for quite a long time transferring even 10Mbits of data. That's about 25 seconds of holding the devices close together. (You can scale that for Gbits...)
...edited to add because of fumble-fingers...
You'd be better off using NFC to bootstrap a more robust, higher-speed connection. Use the NFC as the identification phase of bootstrapping a secure, say, Bluetooth connection or the like.