Bdtyktl

Netgear offer 270Mbps Ethernet-over-coax adapters (MCAB1001) for under $200. They can (apparently) coexist with your cable system (if you have one) and provide a 10/100baseT (RJ45) socket to connect to your LAN/computer.

Not sure what the deal is regarding the advertised 270Mbps transfer speed being limited by the 100bT connection though.

In addition to what others said:
TV coax is not the same as 10Base2 coax. The former has an impedance of 75 Ohms (at least in Europe); the latter has 50 Ohms.

There is a trade body Moca which has this page with a number of ethernet to coax bridges on it. I think this is what you want. However most of them don't look like they are retail products at the moment.

This is not really a serious suggestion...just a dumb, but potentially cool hack.

The cable companies more quite high bandwidth internet activity over 75 ohm coax using only part of the available bandwidth. SO the hardware for this has got to be available, and a standard cable modem is one half of it (and I own two of the things, how about you?). You might even be able to configure several in a peer-to-peer mode with fixed IP address.

Like I said, not serious.

The only option I have seen is a product call TVNet from Coaxsys, but it looks like the product and the company no longer exist. These essentially created an Ethernet transceiver and used some sort of proprietary protocol to communicate over coaxial cable.

Maybe you can find some old units on Ebay.

http://www.hometech.com/techwire/net.html#CX-012000188

There's another alternative that I'm surprised nobody else has mentioned: just install WiFi. If you need multiple connections in one room, you can save on wireless adapters and configure a router running DD-WRT (or your favorite open router firmware) to bridge your wired and wireless networks.

In some houses, it's not easy to run new cable. If that's the case, you can tape the end of your network cable and the end of a rope to the end of the coax in your wiring closet, then pull on the other end of the coax until the network cable and rope are through. To pull the coax back into place, retape just the rope to the coax then go back to the wiring closet and pull until the coax is back. Note that you may or may not have perfect success with this, depending on whether the holes for the cable are big enough to accommodate both coax and network cable, so it might be best to try it first in a room where you'll never need coax (just in case you can't pull the coax back through from the room to the wiring closet).

There's a cheap solution to your problem that nobody has mentioned so far. The ActionTec that are provided by Verizon implement MoCA, which is a standard that allows IP traffic to coexist with CATV traffic on the same coax. (The speed is good - it should be close to 100baseT wired speed). These routers can be configured in bridge mode, so you can use several of them - you'll need one at each endpoint.

These devices (ActionTec MI424WR) can be bought very cheaply on eBay; I've seen them as low as $25/each. So it should be possible to do exactly what you want without spending a fortune and without causing any problems with your TV reception.

More detailed info can be found in these threads:

http://www.avsforum.com/avs-vb/showthread.php?t=1145636

http://www.dslreports.com/forum/r17679150-Howto-make-ActionTec-MI424WR-a-network-bridge

Outside of all these fine suggestions, i'd lurk on AT&T UVerse forums and see what you can dig up.

I just had an install, and the tech tried to use the existing cabling to make my UVerse home net, but we couldn't ever find how to sever from the rest of the building so he just switched to the existing phone networking (forgot to ask if it was cat5).

So, AT&T folks know how to run home ethernet on normal 'cable' coax.

I'm currently facing the same issue; just that the house is rented and thus I have no plan B.

For one, there's even a Wikipedia article about Ethernet over Coax. One "popular" (= easy to stumble upon) option seems to be MoCA. It allows speeds up to 2.5 Gbps ... but the devices that I could find (even the ones for 400Mbps) are quite expensive (starting at 70€, Nov 2018). There's also "HomePNA" and "G.Hn" but the devices seem to be similar expensive.

There's also DOCSIS which AFAIK is used for cable internet (at least in the EU?) but for all that I understand this is absolute overkill for a home installation.

Far cheaper are "DECA" (DirectTV Ethernet to Coax Adapters) ... these are as cheap as 8$ the unit. The speed seems to be limited to 200 Mbps, though.

Another option could be to run a Wifi over Coax ... basically buy 2 Wifi adapters with an external antenna (usually RP-SMA) and instead of using the antenna connect it to your coax cable (e.g. via an adapter RP-SMA to F ... less than 10$ for two usually). Here's an older (2006?) website where someone has done this. I'm not sure, but AFAIK you could probably use more than 2 wifi adapters on a single cable .. using the cable instead of air as transmission medium.

Another more "hackish" solution could be to use Powerline adapters and run them on the Coax cable, similar to how this article describes running Ethernet over a phone cable.