If you don't need traffic on the 10.0.0.0/24 net to pass outside the subnet: Remove the GATEWAY= from the 10.0.0.0/24 interface.
If you DO need traffic to traverse outside the 10net you are looking at more complicated routing tables. Remove the GATEWAY= from both interfaces.
IN: /etc/sysconfig/network-scripts/ifcfg-eth1
add:
DEVICE=eth1
BROADCAST=192.168.0.255
IPADDR=192.168.0.2
NETMASK=255.255.255.0
ONBOOT=yes
IPV6INIT=no
IN: /etc/sysconfig/network-scripts/route-eth1
add:
192.168.0.0/24 dev eth1
default via 192.168.0.1
IN: /etc/sysconfig/network-scripts/ifcfg-eth0
add:
DEVICE=eth0
BROADCAST=10.0.0.255
IPADDR=10.0.0.2
NETMASK=255.255.255.0
ONBOOT=yes
IPV6INIT=no
Now this is the silly part. Let's say that your 10.0.0.1 gateway routes traffic to 10.0.2.0/24 and 192.168.67.0/24 ...We need a static route defined to reach those networks through the proper gateway:
IN /etc/sysconfig/network-scripts/route-eth0
add:
10.0.0.0/24 dev eth0
10.0.2.0/24 via 10.0.0.1
192.168.67.0/24 via 10.0.0.1
I highly doubt you actually are routing outside the 10net on the 10 gateway.. but the above config is the solution for that use case. If you just want LOCAL access to the 10.0.0.0/24 subnet via eth0 do all of the above but replace /etc/sysconfig/network-scripts/route-eth0
10.0.0.0/24 dev eth0
The short version is to remove the GATEWAY line from eth0. Run /etc/init.d/network restart
after making any of these changes.
NOTE: You do not need a gateway if you are not leaving your subnet.