It actually has a lot more processing power than the minimum required to "connect to the Internet" - TCP/IP and web servers have been implemented on microcontrollers with KBs of RAM. Parsing and rendering HTML, however, does need more than that...
> In terms of technology, the achievement is not only in connecting a small computer on the Web, but also in the size of the network software that is running on the chip, according to Shri. The computer consists of an iPic TCP/IP stack running on 256 bytes of memory, using its own equally tiny operating system. Despite the small size, the TCP/IP stack is fully compliant with the requirements of the relevant standards. It is connected to the Internet through a serial port. Because the machine is a Web-server, it does not need a keyboard or display, but is operated from another computer using a Web connection.
...or maybe not, as this C64 web browser(!) shows: http://csdb.dk/release/?id=30400