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: Accompanying live cargo Being an employee of the company flying the cargo
In the remote areas like Alaska it may be possible to be flying a half and half plane.
Half cargo - Half passenger.
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
travel.stackexchange.com/users/1372/Karlson
Most freighter aircraft (excluding smaller aircraft) do have a small number of passenger seats, normally intended for use by staff such as deadheading/additional pilots, other company staff, etc.
Whilst it's possible that freight airlines somewhere in the world do accept paying passengers, in general it's not possible unless you work for the operator of the plane.
Some cargo companies allow select staff to fly on their flights for any purpose, whilst others only allow those travelling for business purposes.
There's a number of reasons that it's not allowed, but in it's simplest form
it's due to the licence class of the aircraft itself, which will not have been certified to carry paying passengers.
In most cases this will be form of the airlines "Air operator's certificate" (AOC) not allowing for passenger traffic, either at all, or on specific types of aircraft such as freighters.
Carrying passengers would also require the flight to have flight attendants in order to meet requirements for safety, and for an aircraft that might have as few as 4 passenger seats that's simply not financially viable!
Even when only carrying their own staff things can still go wrong, such as in the case of Fedex flight 705 .
travel.stackexchange.com/users/984/Doc
Alaska Airlines operates a number of 737-400 "combi" aircraft , which have a cargo area behind the cockpit, and 72 passenger seats in the back.
The airline posts a schedule of which flights will be operated by these planes; they are ticketed just like every other Alaska Airlines flight.
Mostly they are flights within Alaska but a few go to Seattle.
These flights also show up on Kayak as "Boeing 737-400 Mixed Configuration".
travel.stackexchange.com/users/1362/Nate Eldredge
If you're a member of the US military or an immediate relative you can fly Space Available (Space - A).
This can get you on military cargo aircraft (with cargo), some of which are military versions of civilian aircraft. http://www.amc.af.mil/amctravel/spaceatravelcategories.asp
travel.stackexchange.com/users/6541/Will
I worked for Fedex until 2002, and up until September 11, employees could travel on the jumpseats of company planes.
This was only available to employees, and not to friends or family, so when I was single, I traveled quite a bit, even getting to Japan and Hong Kong for free.
Seats were available first come, first served, and at any time you could be bumped off if the company needed to use the seat to move a flight crew or other employee who needed to go where the aircraft was going.
I recall that they suspended this after September 11, and do not know whether or not they have re-instituted it or not since then.
The key was that they were not flying passengers, but allowing employees who had taken a training class on how to operate the emergency equipment on board the aircraft.
travel.stackexchange.com/users/12608/David
Short answer: no.
I have no special knowledge of the subject, so i propose a google potpourri.
"All I know is that I know nothing, except Google" - Socrates
It was the 2012 april's fool for STA travel .
Apparently it was possible for Fedex employees at some point.
Fly on cargo plane Passengers on cargo flights still possible ?
(If you are a pet, it's possible too as there is pet cargo.)
Maybe in case of special merchandise professionals have to stay near (but it's not a travel).
( 3 zoo keepers flew on a cargo because they brought two pandas )
By boat it's feasible, where can I get information on freighter travel? .
travel.stackexchange.com/users/4030/travelot
On special occasions you can not only travel in the normal passenger seats but also inside the cargo hold.
For example in the COVID-19 epidemic Americans went home in a Kalitta B747 freighter like this Click the link on each image for the source
travel.stackexchange.com/users/13777/phuclv
Technically, passenger aircraft are also cargo carrying aircraft, so the answer is yes, however aircraft specifically licensed or designed without passenger seating would only permit flight crews, essential staff that accompany live freight, or in the case of military with armaments.
The liability for passengers and security arrangements would make general passenger transport using freighters a high risk, especially in the current situation of heightened terrorist awareness.
An open approach would be counter to these measures.
travel.stackexchange.com/users/18657/Elwyn Jones
Some time ago I heard someone telling about travelling in cargo planes. Apparently some of those planes have a limited amount of extra seats and companies would sell them for an extra profit.  Does anyone have any information about this? Is that still possible?
Is it possible to travel on cargo airplanes?
travel.stackexchange.com
<air-travel><freighter-travel>
travel.stackexchange.com/questions/14297
travel.stackexchange.com/users/3736/nsn
In remote areas like Alaska it may be possible to be flying a half and half plane, but in general it is not possible with a few exceptions such as   accompanying live cargo, or being an employee of the company flying the cargo. Some cargo companies allow select staff to use their jumpseat service for any purpose, whilst others only allow those travelling for business purposes. Aircraft licenced to fly cargo would normally not be permitted to have paying passengers, or doing this would require extra staff and safety standards tha would not be financially viable. Air Alaska runs a half cargo half passenger plane service but that is very rare. Most cargo companies only allow crew and employees to travel on cargo planes, and nowadays, even that is very rare. Regulations state that planes carrying passengers must have a cabin crew.
It is generally not possible to travel on cargo planes, unless you are an employee or crew member. It used to be allowed for employees by Fedex for example but not anymore. Air Alaska has half cargo half passenger planes. Tickets are sold as normal airline tickets. Technically a plane can carry passengers but, in order to do so, it would need a cabin crew to meet safety regulations.