For an object in orbit to remain at the same position relative to a point on the ground, it needs to be in geostationary orbit, i.e. at an altitude of 35,786 km, and directly over the equator.
That orbit is quite densely occupied by a large number of satellites, so they would have to find an available spot.
That orbit is also a much higher orbit that that used for the usual manned missions, which are all in Low Earth Orbits, about a hundred times closer to Earth. Getting there requires a higher delta-v budget, about 40% more than for LEO, so a much bigger rocket (to get more delta-v you need more fuel so the rocket is heavier so you need more fuel so...).
The delta-v requirements are similar to those to get to Lunar Orbit, so you know it’s definitely possible. We no longer have the mighty Saturn V which took astronauts to the Moon, but see Edit #2 below for alternatives.
The question is then: what would be the purpose of such a mission?
To clarify, a spacecraft in another orbit cannot just "stop" and remain above the same spot on the ground. For each altitude, there is a speed to be maintained, which is just the right balance between inertia (which would make the spacecraft continue in a straight line and move away from Earth) and gravity (which tries to pull the spacecraft down towards the Earth).
If the spacecraft slows down, the balance is altered, and gravity will slowly pull down the spacecraft towards Earth. It's all very well illustrated by Newton's Cannonball:
Geostationary orbit is just the orbit where that speed matches the rotation of the Earth.
A small capsule for a short mission like the Gemini capsule weighed less than 4000 kg. The Atlas V 551 is powerful enough to lift that to GEO, and the following (currently operational) other rockets are more capable: Ariane 5 ECA, Long March 5, Delta IV Heavy, and Falcon Heavy (expended).
The Falcon Heavy is over 3 times as powerful as the Atlas V 551, so even a larger spacecraft would be quite possible. You wouldn't get something as big (and heavy) as a Space Shuttle orbiter up there, though (by far).
Note that there are quite a few even more powerful rockets in development.