Can I use DOCSIS 3.0 to connect to my LAN over coax?
You need an ethernet-over-coax device for RG-6 cable.

A slightly surprising source of these is CCTV suppliers, as there is a lot of analogue CCTV wiring out there, and people want to upgrade to higher resolution IP CCTV, without running new cable. Therefore, converters to run Ethernet over Coax are available. 

Example


  VHW-HW    HIGHWIRE Ethernet over coax converter (2 required per link)
  100BaseT up to 350m (1150ft) full-rate over RG-6


https://www.veracityglobal.com/products/ethernet-over-coax-devices/highwire.aspx

One supplier

http://www.eaccu-tech.com/by-vendor/vhw-hw-veracity/

There is also a possible hack solution - connect wifi antennae connectors directly to the co-ax cable. 


  There are some problems with this plan. WiFi is 50 Ohms, TV coax cable
  is 75 Ohms. Only one MIMO channel will be available meaning the
  maximum theoretical bandwidth will be 433 Mbps. WiFi is also at much
  higher frequencies than what coax is designed for.
  
  With two WiFi antenna to coax adapters, [Tobias] simply connected the
  coax directly to a router set up to bridge Ethernet over WiFi. The
  entire thing worked, although testing showed it was only getting about
  60 Mbps of throughput.


https://hackaday.com/2018/10/19/wired-wireless-over-coax/