The general answer is: depends.

There are multiple discussions on Airliners regarding the subject. One from 2004 and one from 2011.

General consensus is that it is not possible with few exceptions:

1. Accompanying live cargo
2. Being an employee of the company flying the cargo
3. In the remote areas like Alaska it may be possible to be flying a half and half plane. Half cargo - Half passenger.
4. You are an airline crew member or a courier in that case you can request a Jumpseat Service from cargo airlines like Atlas (maybe someone can enlighten me what Groom means here)

Save those you are likely to be denied access to the plane

The short answer is: depends.

There are multiple discussions on Airliners regarding the subject. One from 2004 and another from 2011.

General consensus is that it is not possible with few exceptions:

1. Accompanying live cargo
2. Being an employee of the company flying the cargo
3. In the remote areas like Alaska it may be possible to be flying a half and half plane. Half cargo - Half passenger.
4. You are an airline crew member or a courier in that case you can request a Jumpseat Service from cargo airlines like Atlas (maybe someone can enlighten me what Groom means here)

Save those you are likely to be denied access to the plane