Ethernet to serial is something I'm actively working on at the moment as part of the great wireless mesh project. It is not a simple thing, mainly because it is expensive or complicated or both. That device above is $59. The absolute cheapest method may be a propeller chip ($7.99) and an enc28j60 ($4). But I don't understand propeller code. I've gone for an option using lantronix devices. The xport is probably the best known ($47), but on ebay I managed to pick up three MSS100 devices for $60 including shipping so that worked out at $20 each. They come up from time to time, and indeed there are lots of other lantronix parts there that all work in a similar way. Search for lantronix on ebay. Then there are devices called 'terminal servers' or similar. Also made by lantronix, they look like router boxes and even have the same connector at the front, except the connector is actually a serial port, not an ethernet port. So one ethernet cable goes in and you can have 16 serial ports coming out and the cost per port ends up being very cheap.
The MSS100 has taken a group of five people about two weeks to get working, but we now have it such that you can log in via telnet from anywhere in the world and send data to and from a cp/m board or, indeed, a picaxe chip. We are close to getting the 'dial out' working too.
What is needed is a device that costs under $10 and is a bit easier to program than the lantronix devices and can interface an ethernet cable to a serial link. Program in the IP address, baud rate, parity, stop bits, a welcome login text line, timeout parameter and a few other things. Then do a portforward on your router to open up that port.
The "poor man's" version is to use a PC as the ethernet to serial device and an old (freebie) PC can be pressed into service. But this uses a lot more power and takes up a lot more desk space.
Maybe someone has some ideas for getting a picaxe onto the internet for a good price. Stelios did it by hacking a router and that might be another option.