How can I tunnel all of my network traffic through SSH?

To do what you are wanting, I recommend sshuttle.

You use it like this:

./sshuttle -r username@sshserver 0.0.0.0/0 -vv

It will tunnel all your TCP traffic automatically for you. You can add the --dns argument to have it tunnel your DNS traffic as well. The remote server only needs to have Python installed.

If you only want to tunnel specific programs I would recommend proxychains.

Once it is installed, start your ssh socks proxy like this:

ssh -fNTD 127.0.0.1:<local port> username@sshserver

This will start a "SOCKS" proxy listening on <local port>.

Then edit /etc/proxychains.conf to point to the same port as <local port>:

socks5 127.0.0.1 <localport>

Finally start your program that you want proxy-ed like so:

proxychains <program name>

It should just work. However, a few programs will have trouble working with Proxy Chains. Also keep in mind, that with Firefox, you have to change additional items under about:config to force it to do DNS lookups through the proxy instead of bypassing it.

As an additional note, on web browsers. If they support socks proxies, you don't need to do anything additional to get them to use the above mentioned, ssh tunnel, just enter 127.0.0.1 for the SOCKS proxy server and the <local port> for the proxy port.

EDIT 3/29/16

Since this post is still seeing some upvotes, I thought I'd update it. Proxychains is still in most Linux repos and still works on Linux. However, the project is effectively abandoned and does not work on OSX. For either Linux or OSX, I highly recommend upgrading to a still-maintained fork: proxychains-ng: https://github.com/rofl0r/proxychains-ng

Besides working in both Linux and OSX, it is easy to compile, and also has much better support for DNS tunneling.

I should also mention another option, which is redsocks. It works similarly to proxychains(-ng) and is also likely in your dist repo: https://github.com/darkk/redsocks

EDIT 11/27/19 If you go the proxychains route, please use proxychains-ng. There are some serious bug fixes over the legacy version, like: https://github.com/rofl0r/proxychains-ng/issues/292

man ssh gives an example of exactly this. An ssh based vpn:

SSH-BASED VIRTUAL PRIVATE NETWORKS
     ssh contains support for Virtual Private Network (VPN) tunnelling using
     the tun(4) network pseudo-device, allowing two networks to be joined
     securely.  The sshd_config(5) configuration option PermitTunnel controls
     whether the server supports this, and at what level (layer 2 or 3 traf-
     fic).

     The following example would connect client network 10.0.50.0/24 with
     remote network 10.0.99.0/24, provided that the SSH server running on the
     gateway to the remote network, at 192.168.1.15, allows it:

       # ssh -f -w 0:1 192.168.1.15 true
       # ifconfig tun0 10.0.50.1 10.0.99.1 netmask 255.255.255.252

~~ snip ~~

     Since a SSH-based setup entails a fair amount of overhead, it may be more
     suited to temporary setups, such as for wireless VPNs.  More permanent
     VPNs are better provided by tools such as ipsecctl(8) and isakmpd(8).

Once you have that new interface up, you'd just have to make it the default route, which is a different question.

Look for the "Tunnel" option in ssh. This creates a tunnel device that you can assign an IP address to, and then you change the default route to use that tunnel.