The human will die instantly if blown out of the cannon
(Note: skip to "Launch Detection" for how to address the changed focus of the question.)
A back of the envelope (literally) calculation full of approximations shows that in order for the bullet to reach the mesosphere (50 km ASL) on a ballistic arc, launched from sea level, it needs to be launched at around 1200 metres per second. Note that this will just reach an altitude of 50 km after 80-90 seconds and start falling back, the rocket will need to do all the work of getting anywhere after this.
Assuming perfectly smooth acceleration over a barrel length of 300 metres (1000 ft), acceleration will be on the order of 2300 metres per second squared for almost half a second. While all my numbers could be off by as much as 50% in either direction (lots of approximations) 230 G is simply not survivable. Even by adding all plausible (and some implausible) favourable factors, humans can't get into space this way in any condition to achieve a mission.
Single stage to orbit - not that easy...
The main difficulty in getting into orbit is not achieving the required altitude but reaching the required orbital velocity of over 9.4 km/sec. Escape velocity is 11.2 km/sec. This is not practical with current technologies (which is why multi-stage rockets are used to get to orbit in the real world).
Is there an alternative?
All you are launching is a single soldier in exoskeleton armour. This doesn't look like a long range or long duration mission, more like a special forces job in earth orbit. In that case - why send a fragile human? Send a drone (or 3 smaller ones), controlled from groundside. The drone will need some pre-programmed routines, but lightspeed delay is minimal, especially if in low-earth orbit. (Although you will need to keep switching to a new retransmission station with line of sight as the drone orbits.) This allows you to use your payload for mission-critical capabilities instead of life support and means you no longer need to worry about how to retrieve the operator at the end of the mission. However, given that you have clarified that the crucial element of the story is to deliver four humans into space unnoticed, the following section will examine how to do this.
Launch detection and covert options
Assumptions regarding the enemy, designated Red Force:
1. Red force has finite forces that are in total control of Earth orbital space.
2. Red force has satellites in orbit with ground observation capabilities sensors equivalent to current United States surveillance satellites.
3. Red force has other sensor capabilities equivalent to current United States capabilities.
4. Red force has both space-to-surface and space-to-space weaponry equivalent to current Earth prototypes or near-future concepts, primarily mass drivers or missiles.
As other answers have covered, it would require a linear accelerator hundreds of kilometres long to launch a person into orbit without fatal acceleration levels. Building such a linear accelerator (overcoming massive engineering challenges) without being detected and attacked from orbit would be extremely difficult, time consuming and hideously expensive. So let's look at other options using conventional launch vehicles.
If a satellite has its sensors watching an area, there is no real way to disguise a missile or spacecraft launch. Any currently existing rocket which is efficient enough to get a payload into orbit will produce a massive thermal signature. It's the same as being in a very dark room, yet the tiny indicator lights on appliances are easily visible. The only way to mask them is to put something thermally opaque between the sensor and the thermal source (not practical), increase the temperature of the entire region to match the temperature of the rocket exhaust (not survivable), wait for the observer to stop watching (maybe) or poke out the eyes of the observer (ahhhh!).
The Defenders need to be tracking the enemy satellites in order to make this work - no alternative. If they are lucky there is an existing hole in the Red Force satellite coverage that they can take advantage of, somewhere that is not being directly observed for a few minutes every so often. If such locations do exist then they are probably in bad locations for launches - it would be all too easy to keep the entire equatorial area under observation, for example. If there is no practical location that they can take advantage of then they need to get into the anti-satellite business using either rockets, cannon or lasers.
So there is a use for the cannon - a bunch of depleted uranium marbles in a casing can survive a savage acceleration and only need to reach the satellite's altitude for a moment - then the satellite's > 9km/s speed will provide all the kinetic energy necessary to destroy it. Ditto for a rocket, no great performance required. Regarding lasers, I am suggesting tracking the satellite with a laser powerful enough to dazzle the onboard sensors, not something powerful enough to cause material damage. If at all possible, hit additional satellites covering different areas at the same time, otherwise Red Force can focus all its reserve sensors on the one area.
While the eyes are poked out or looking elsewhere is the time to launch. All four humans launch in a single 2-stage vehicle with a bullet-shaped final stage. At roughly the same time five smaller rockets also launch. The smaller rockets will deliberately have the same acceleration profile as the primary and cube-corner radar reflectors to give them the same radar signature. Once they are out of the atmosphere they will inflate "caps" at the front to give them the same visual profile as well. (This is assuming that the Defenders are on a budget and cannot afford to waste six full-size launch vehicles.)
Now switch to Red Force's viewpoint for a while. They just lost a few satellites or their imaging, refocus other sensors on the area and pick up six bogeys as they clear the atmosphere. It looks like the Defenders had trouble getting quality components because one suffers a catastrophic failure at first-stage separation. The other five continue on but as they get close to an intercept course on an important Red Force target two more malfunction - the lead one explodes into tiny fragments and the last one loses thrust and starts venting fuel on one side, throwing it into a flat spin. The three remaining rockets accelerate towards the Red Force target to ram it but miss, either as a result of more poor engineering or point defence fire. Red Force has emerged victorious, albeit fairly easily against the ragtag Earthlings...
Switching back to the Defenders' viewpoint - the rocket that failed at first stage separation was to make the subsequent malfunctions believable. The second rocket to go will explode into prefragmented pieces designed to provide maximum visual and radar distractions, with the direction of shrapnel release to provide masking without endangering the humans. The four humans were aboard the third rocket - the rocket's spin was carefully planned to be in a plane that will give them the vector they need to reach their target (or get close enough for low-impulse thrusters to do the rest) if they release at the correct point in the rotation. The humans' armour is as non-reflective to visual and radar wavelengths as possible and includes thermal masking - I strongly recommend it is made of carbon composites as much as possible instead of depleted uranium. Provided they keep tight emissions control, the humans will look far less interesting and important than the thousands of bits of space debris in the area.
I acknowledge the influence of the late Nigel D Findley in the plan presented above - see p127 of the Aztlan sourcebook for Shadowrun 2nd Ed.
Edit 1 - previous alternative
Recent events in Hawai'i have inspired an enhanced solution, albeit with no research to back it up. While it would be a really bad idea due to the geological instability of the area, let's posit an undersea tunnel linking O'ahu to the Island of Hawai'i. Hidden from prying orbital eyes, this tunnel could be repurposed into a linear accelerator a few hundred kilometres long, with the normal exit on the Island of Hawai'i extended and aimed upwards, albeit in a gentle curve. There are lots of engineering difficulties, such as evacuating all the air in the tunnel in front of the bullet, but these are not inherently impossible if the tunnel has sufficient integrity. Time the launch to coincide with a volcanic eruption (predicted or induced) and the Defenders can launch their projectile with hot ash covering any thermal signature and the eruption covering for the shockwave.
I realise this is nowhere near the original specifications of the OP's cannon. However, it does give a vaguely plausible mechanism for an undetected, human-survivable launch, with the added drama of requiring precise timing to avoid the bullet being destroyed by airborne debris from the eruption.