Is an "atmospheric geostationary satellite" feasible with current technology?
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.