Meaning by "atmospheric geostationary satellite" a vehicle capable of hovering 30 km above Earth surface, hence insde atmosphere, for unlimited time, making use of air propellers, solar panels and batteries.
How much energy is needed to keep an object steady in air at 30 km above surface?
There are two possibilities: a plane-like vehicle, flying in circle, using wings to stay up; an helicopter-like vehicle, hovering.
Short answer: No.
Long answer: Your "unlimited time" condition makes the short answer easy. If you need to go up for a limited time, and you have the liberty to choose the time of ascent and descent, the answer is: Possibly.
First, there are two ways of going up there: A balloon and an aircraft. A helicopter would be much less efficient, the tether of a balloon will be too heavy to lift, and you will see that even a glider-like aircraft will struggle to stay for some time at 30 km.
Next, atmospheric pressure at 30 km is 12 mbar, just 1.2% of what it is on the ground. So your aircraft has to go at some speed just to create enough lift. If we assume that you fly at Mach 0.5 to 0.6 (where high-lift airfoils and propellers can still operate), your dynamic pressure is that of a modern glider at low speed. The flight speed actually is between 150 and 180 m/s, however.
Now let's assume your aircraft has an L/D of 50. This translates into a power requirement just to stay aloft of 3 to 3.6 kW per ton of aircraft mass. Your propeller will have no more than 85% efficiency, and your engine will create some loss as well, so your installed power needs to be at least 4 kW per ton of aircraft mass. Since you fly at a glider's dynamic pressure, you will also have a glider-like wing loading of 30 kg/m². Per ton, your wing area is 33.3 m². To operate an air breathing engine at this altitude is a challenge, so I will assume solar electric propulsion. Let's be optimistic and we assume a solar constant of 1.4 kW/m² at this altitude, and you fly at moderate latitudes (say, 45°), so your solar panels can create (20% efficiency) 200 W per m² or 5.3 kW per ton of aircraft mass (assuming 80% of the wing is covered, the rest will have too much curvature). Of course, this is all valid only around noon, so if you want to stay overnight, even in summer there is not enough energy to keep the aircraft up for a longer time, even with very optimistic assumptions. You might want to assume batteries which can provide the power for the climb phase, so you take off in the middle of the night and arrive (with empty batteries) at altitude in the late morning. Then you might really stay there until 2 or 3 pm, when the lowering sun means that your solar panels will not provide enough power to stay up, let alone charge the batteries. And I have not even started to subtract the power for operating any equipment (which would be the reason to go up there in the first place, right?).
To go with a lower wing loading means you will have a very delicate structure which cannot cope with the winds at 30 km and gust loading at lower altitude.
This wind will also be a problem for any ballon. 20% of winds at 30 km are above 76 m/s (Mil Std. 210C), 5% are even above 98 m/s. One cubic meter of helium gives you 10 N of lift at sea level, but only 0.156 N at 30 km. To lift one ton of balloon up to 30 km, you will need 63,000 m³ of volume. This is a sphere with a diameter of 50 m! The wind forces to keep this thing tethered will need a strong and heavy tether, so without going into the details I think that a tethered balloon will be a challenge, to put it mildly.
The "Facebook drone" articles are not serious, they are meant to create some P.R., and don't hold your breath for Internet delivered by drones anytime soon.
In 2003 NASA's Helios solar-powered aircraft made it up to 29.52 km (rounds up to 30 km!). Alas, it broke up in flight, and I can find no reference of a follow on. A plane would be more energy efficient than a helicopter, even a helicopter in forward motion, so I expect a plane is what you would see first. The problem of storing energy and flying at night may be insurmountable. (Perhaps beaming microwave energy to the aircraft?)
You could almost imagine a blimp or dirigible with propulsion to fight the wind, but it would be difficult to get much payload at that altitude, and the wind up there is fast. So, no way.
If you don't mind a long line that aircraft would have to avoid, a kite is another possibility, and would solve the problem of power for the payload at night. There are pretty constant winds up there, so if you can get it up, you can imagine a kite flying for a long time.
A tethered balloon could in principle work as well, and would have more payload than a free balloon trying to fly against the wind. However the balloon envelope would have to be strong, and therefore heavy, to withstand being held against the wind by the tether. Even a balloon envelope only three times as thick as those designed for zero-wind conditions would result in no payload capacity at all.
The kite seems the most plausible, but that altitude is well beyond the current record of 5.7 km. A series of kites along the line can help carry the mass of the tether.
None of these things could be referred to as a "satellite".
Depending on your definition of "unlimited" it is not only already possible but you can (or could) already buy it.
Titan Aerospace introduced the Solara last year. It's a solar powered plane with an expected airborne time of 5 years. Obviously they haven't tested it for 5 years but worst case is that you'd have to land it every couple of years for maintenance. The target market is specifically atmospheric satellite.
They had a web page up where you could order one. Last time I checked their website implied that they had 5 orders. They had a progress indicator of when you can expect to get your plane if you have several million dollars to spend and they said that it takes about a month to build one. When I checked they said the earliest you can expect it was 5 months away.
Unfortunately since Google bought the company the website's been down so there's nothing much on their site. But if you google "Titan Solara" you may see pictures of their plane.
Here's an Arstechnica article about the Solara: http://arstechnica.com/information-technology/2013/08/almost-orbital-solar-powered-drone-offered-as-atmospheric-satellite/
Note: The Solara is designed to loiter around 20km up, not 30km.
Facebook is considering building solar powered communication drones. According to the linked Verge story, they'd hover at 20 km.
I wouldn't call these drones satellites, they're not in orbit. An orbit at that altitude would have a speed of nearly 8 km/s and the period of the orbit would be about an hour and a half. A low altitude orbital sat wouldn't hover stationary over a given location. You need to get up to about 36,000 kilometers before the sats slow down enough to match the earth's rotation.
Edit: adding an illustration to show how altitude affects radius of footprint.
Where r is radius of earth (6378 km) and a is altitude drone (either 20 or 30 km).
Footprint radius for low altitudes is approximated by r sin α where α is acos(r/(r+a))
For a drone at 20 km this comes a radius of 503 km, for a drone at 30 km, this comes to a radius of 616 km. (503/616)^2 is ~.668. Lowering drone to 20 km would give a footprint of about 2/3 the area.
